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CURVES 


OF 

CIVIL  ENGINEERING. 

THE    TRANSITMAN'S    POCKET 
COMPANION. 

CONTAINING  INSTRUCTIONS  FOR  SURVEYS  TO  LOCATE 

THE  CENTER  LINE  OF  ENGINEERING  WORKS, 

AND  TABLES  THAT  ARE  NEEDED 

IN   THE   FIELD, 

IN    CONNECTION    WITH 

CIRCULAR,  TRANSITION,  AND  VERTICAL  CURVES. 


PRICE,    $2.50. 


ARTHUR  M.   HAYNES,   C.E. 

M.   iv'.'s.  £. 


37  So.   LINCOLN  AVE.,  DENVER,  COLO. 


1903. 


- 


y 


Copyright,   1903, 

by 
ARTHUR  M.  HAYNES. 


ROBERT   DRUMMOND,    PRINTER,    NEW    YORK. 


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One  writer  has  furnished  us  a  table  of  corrections  to 
apply  to  the  more  important  functions,  but  it  requires 
.interpolation,    and  is   confusing  when  taken  with   the 
distractions  of  field-work.     Another  has  given  us  volu- 
minous tables  for  curve  functions,  for  curves  of  integral 

117655  3 


ROBERT   DRUMMOND,    PRINTER,    NEW    YORK. 


PREFACE. 


OUR  forefathers  began  laying  out  curves  with  loo-ft. 
chords,  because  it  was  a  little  easier  to  do  field-work 
that  way.  In  their  practice  the  errors  were  not  so  seri- 
ous as  they  now  are,  and  their  link  chains  made  it  more 
difficult  to  measure  arcs  (or  anything  else)  than  with 
steel  tapes  now  used. 

We  have  followed  their  example  too  long.  This  prac- 
tice is  responsible  for  countless  shocking  kinks  in  rail- 
road curves  that  are  covered  up,  whitewashed,  and 
unknown.  The  average  transit  party  does  not  expect 
to  close  on  the  P.  T.  closely  when  it  had  been  previously 
set  according  to  tables.  No  matter  how  carefully  they 
may  do  their  work  they  know  there  are  errors  in  the 
tables  that  will  make  fudging  necessary.  The  method 
is  known  to  be  inaccurate,  and  is  made  the  excuse  for 
careless  work.  The  errors  have  been  characterized  as 
"inappreciable"  or  "negligible,"  but  they  are  cumula- 
tive and  increase  as  the  square  of  the  degree ;  have  made 
much  trouble,  and  have  thrown  discredit  upon  the  pro- 
fession. 

One  writer  has  furnished  us  a  table  of  corrections  to 
apply  to  the  more  important  functions,  but  it  requires 
interpolation,  and  is  confusing  when  taken  with  the 
distractions  of  field-work.  Another  has  given  us  volu- 
minous tables  for  curve  functions,  for  curves  of  integral 

117655  3 


4  PREFACE. 

degrees  from  i°  to  10°.  That  work  (Butts)  has  been 
highly  appreciated  and  ought  to  be  in  the  hands  of 
every  conscientious  transitman,  but  it  is  limited  to  the 
integral  degrees  i°  to  10°,  and  only  demonstrates  the 
impracticability  of  chord  measurements.  The  mere 
idea  of  measuring  across  a  chord  100  feet  and  calling  it 
the  length  of  the  curve  is  revolting.  The  error  for  a 
20°  curve  is  a  half  of  a  foot.  It  might  be  excusable  if  it 
resulted  in  ease  and  simplicity  in  doing  the  work.  But 
it  results  in  difficulties  and  uncertainties.  As  one  indi- 
cation of  the  difficulties,  there  is  not  a  curve  shown  on 
any  of  the  filed  plats  of  surveys  for  irrigation  canals  in 
Colorado.  They  aggregate  10,000  miles;  some  of  them 
cost  a  half  million  dollars,  and  are  fine  examples  of 
engineering,  but  their  alinements  are  all  attempted  to 
be  shown  by  straight  lines  and  angles  because  chord 
measuring  cannot  be  used  on  the  sharp  curves.  The 
excuse  for  this  book  is  to  advocate  arc  measurements. 

A  new  table  of  i°  Curve  Functions  is  here  given,  in 
accordance  with  a  new  definition  for  "Degree  of  Curve," 
which  is  found  in  the  first  paragraph  of  the  text.  With 
measurements  on  the  curve  the  table  is  exact  for  all 
degrees.  With  measurements  on  chords,  results  are  as 
close  with  this  table  as  with  similar  published  tables. 
So  this  table  is  prefectly  applicable  on  work  where  the 
errors  of  chord  measuring  is  considered  negligible. 
Logarithms  and  tables  of  natural  functions  are  not 
given  because  they  should  not  be  used  in  the  field. 
Transit  work  is,  or  should  now  be,  in  trained  hands, 
which  are  supplied  with  such  tables  as  those  published 
by  Professor  Jones,  of  Cornell  University,  by  Von  Vega, 
or  by  the  U.  S.  Geo.  Survey.  Special  problems,  too,  are 
considered  as  belonging  in  the  classroom. 

The  subject  of  transition  curves  is  made  a  prominent 
feature.  The  discussion  is  much  longer  than  necessary, 
because  there  has  been  so  much  literature  on  the  sub- 


PREFACE.  5 

ject  which  has  to  be  referred  to.     There  seems  to  be  no 
other  subject  upon  which  engineers  are  so  divided. 

A  section  on  vertical  curves,  too,  is  added,  so  that 
the  ground  may  be  better  covered,  as  indicated  in  the 
title.  While  this  subject  applies  especially  to  railroads, 
it  should  be  applied  to  city  street  grades  much  more 
than  it  is.  Street  cars  are  unable  to  follow  safely  the 
grades  that  are  established  by  the  city  engineer,  and 
the  unsightly  appearance  of  sharp  angles  in  street  grades 
is  discreditable. 

The  engineer's  practice  is  of  two  kinds:  one  of  loca- 
tion, the  other  of  construction;  one  of  planning,  the 
other  of  executing;  one  dealing  only  with  nature,  the 
other  with  men,  manufactured  materials,  and  machines. 
This  work  is  to  assist  in  the  first-named  practice ;  in 
the  location  of  a  center  or  base  line,  both  horizontally 
and  vertically,  from  which  all  detail  plans  and  measure- 
ments may  be  made.  A  line  which  will  pay  due  respect 
to  the  inertia  of  moving  objects  and  to  civic  aesthetics, 
thus  to  ''make  the  dollars  earn  more  interest."  For 
nature  abhors  an  angle. 

A.  M.  HAYNES. 

DENVER,  COLO.,  October,  1903. 


CONTENTS. 


CHAPTER  I. 

CIRCULAR   CURVES. 

PAGE 

1 .  Definition  of  degree  of  curvature 9 

2.  Method  of  projecting  curves  on  the  ground 9 

3.  General  definitions 10 

CHAPTER  II. 

ALINEMENTS    OF   ENGINEERING    WORKS. 

4.  Method  of  staking  out 10 

5.  Transit  points  and  plusses 1 1 

6.  Markings  on  stakes 1 1 

CHAPTER  III. 

LINEAR  MEASUREMENTS. 

7.  Tools 12 

8.  Manner  of  using 12 

9.  Slope  measuring 13 

10.  Curve  measuring 13 

CHAPTER  IV. 

GENERAL  DISCIPLINE. 

11.  Rules  of  conduct 14 

12.  Orders  few  and  considerate 14 

CHAPTER  V. 

ANGULAR  MEASUREMENTS. 

13.  Transitmen  should  be  relieved  from  discipline  and  manage- 

ment       15 

14.  Care  to  be  taken 15 

CHAPTER  VI. 

TRUE   MERIDIAN. 

15.  To  obtain  observations  upon  Polaris 16 

CHAPTER  VII. 

FIELD   WORK. 

16.  To  stake  out  curves  connecting  tangents 17 

17.  Definitions  of  functions  of  curves 17 

18.  Plan  of  operations  where  curves  predominate 18 

7 


8  CONTENTS. 

CHAPTER  VIII. 

TEANSITION   CURVES. 

19.  Requirements  for  a  system 19 

20.  Description  to  meet  requirements 19 

21.  Selection  of  suitable  curves 21 

22.  Deflection  angles  for  spirals 22 

23.  Elements  of  spirals 23 

24.  Description  not  needed  in  practice 23 

25.  General  equation  and  formulas 24 

26.  General  deflection  table „.  —  25 

CHAPTER  IX. 

PLATTING. 

27.  Duties  of  transitmen  regarding  same 26 

28.  Order  of  platting 27 

29.  Compound  curves,  special  treatment 27 

CHAPTER  X. 

TURNOUT   CURVES. 

30.  Description 28 

31.  Complications  to  avoid 29 

CHAPTER  XI. 

VERTICAL   CURVES. 

32.  Definitions  and  description 30 

33.  Compensation  for  curves  on  grades 32 

34.  Vertical  curves  too  little  used 32 

TABLES  AND  DIAGRAMS. 

Table       I.  Differences  in  lengths  of  Arcs  and  Chords 34 

II.  Diagram.     Sample  transit  notes 35 

III.          "  Superelevation  of  Outer  Rail 30 

"        IV.          "  Corrections  in  Chaining 40 

V.  Emergency  Table  for  Determining  Natural  Func- 
tions of  angles 41 

"         VI.  Curve  Characteristics 43 

"      VII.  Ordinates  for  Vertical  Curves 44 

"     VIII.  Azimuth  of  Polaris  when  at  Elongation 44 

"        IX.  Functions  of  a  i°  Curve 45 


CURVES  OF  CIVIL  ENGINEERING. 


CHAPTER 

CIRCULAR   CURVES. 


1.  The  Degree  of  a  Curve  is  a  term  applied  to  circitlar 
curves  in  a  plane  and  is  the  change  of  direction  in  degrees 
per  one  hundred  feet;  or,  in  other  words,  is  the  number 
of  degrees  in  an  arc  of  the  curve  one  hundred  feet  long. 

The  radius  of  a  curve        =  572Q'5   . 

degree 

The  radius  of  a  i°  curve  =—     —  =  5729.57795. 

7T 

2.  Circular  curves   are  projected  on   maps  by  means 
of    their   radii    with    the    compass.      On    the    ground 
it  is  generally  found  necessary  to  make  use  of  the  geo- 
metrical theorem   that   an   arc   subtends   an   angle   on 
the  circumference  equal  to  one  half  the  angle  at  the 
center  subtended  by  the  same  arc.     A  transit  is  placed  on 
the  curve  (or  circumference)  and  the  curve  is  located  in 
loo-ft.  arcs  by  turning  angles  for  each  arc  equal  to  one 
half  the  degree  of  the  curve.  The  work  can  be  commenced 
and  carried  on  at  an  unknown  distance  from  the  transit, 
but  it  is  generally  required  to  begin  at  the  transit  so  that 
it   may   conform   with   a   previously   located   tangent. 

9 


IO  CURVES    OF    CIVIL    ENGINEERING. 

When  there  is  special  reason  for  doing  so,  arcs  of  less 
than  TOO  ft.  can  be  located  by  deflecting  angles  pro- 
portional to  the  length  of  arc  measured. 

It  is  bad  practice  to  try  to  locate  points  on  a  curve 
more  than  60°  from  the  transit,  because  one  end  of 
the  tape  line  is  held  on  the  curve,  while  the  other  end 
describes  an  arc  .with  the  tape  as  a  radius,  and  the  new 
point  being  fixed  on  the  curve,  is  at  the  intersection  of 
this  arc  with  the  line  of  colimation  from  the  transit. 
This  angle  of  intersection  is  o°  at  180°  from  transit,  and 
is  too  acute  for  accuracy  beyond  60°  from  transit. 

Tt  is  impracticable  to  have  the  measuring  line  on  the 
curve.  So  it  is  placed  on  the  chord  in  a  straight  line, 
but  the  end  is  held  back,  distances  given  in  Table  I, 
so  that  the  result  is  the  same  as  though  actually 
measured  on  the  arc. 

3.  A  curve  of  uniform  degree  in  one  direction  is  a 
Simple  Curve.     Two  or  more  curves  in  the  same  direction 
connected,  and  with  a  common  tangent  at  point  of  con- 
nection, is  a  Compound  Curve.     If   such   curves   are   in 
opposite  directions  they  form  a  Reverse  Curve.     A  Tangent 
to  a  curve  is  a  straight  line  which  intersects  it  but  does 
not  cross. 

CHAPTER  II. 

ALINEMENT  OF   ENGINEERING  WORKS. 

4.  Engineering    works,    such     as     railways,     canals, 
boulevards,    dams,  tramways,   etc.,    are    surveyed   and 
staked   out   with    a   line   of   stakes    one   hundred  feet 
apart    and    numbered    consecutively,    beginning    with 
station  0  at  one  end  of  the  work.     Such  points   are 
called   stations;    when  it  is   necessary  to  have  points 
between   two   stations  the   distance  is  measured  from 
the  last  station  and  this  distance  is  called  a  plus.     The 


CURVES    OF    CIVIL    ENGINEERING.  I  I 

stake  is  marked  the  last  station   +   the  distance  from 
it  in  feet  and  tenths  thereof. 

5.  At  station  0  and  at  other  points  along  the  line 
where  it  is  necessary  to  set  up  the  transit,  a  hub  is 
driven  flush  with  the  ground  and  a  tack  put  in  the  hub 
to  mark  the  exact  point.     The  stake  is  driven  in  the 
ground  vertically  one  foot  left,  or  if  to  be  on  a  curve, 
on  the  convex  side  of  the  curve  and  facing  the  hub. 
Such  points  are  called  transit  points,  and  the  stakes  to 
the  side  are  witness  stakes,  which  have  markings  on  the 
back  side  as  follows: 

6.  P.  O.  T.,  for  a  point  on  tangent. 

P.  R.  C.,  for  a  point  of  reverse  curve. 

P.  R.  S.,  for  a  point  of  reverse  spiral. 

P.  C.  C.,  for  a  point  of  compound  curve. 

P.  O.  C.,  for  a  point  on  curve. 

P.  S.,  for  a  point  at  beginning  of  a  spiral. 

P.  C.,  for  a  point  at  beginning  of  a  circular  curve. 

P.  T.,  for  a  point  at  beginning  of  a  tangent. 

P.  I.,  for  a  point  at  intersection  of  tangents. 

Sometimes  the  degree  of  the  curves  and  the  whole 
alinemerit  is  indicated  on  the  stakes  from  which  any 
one  can  take  notes  and  plat  the  whole  line.  It  is  not 
advisable  to  give  to  the  public  such  information,  and 
friends  of  the  enterprise  do  not  need  transit  notes  in 
that  form.  Until  late  years  such  alinements  have  con- 
sisted only  of  circular  curves  and  tangents.  But  the 
refinements  of  railway  operation  now  demand  that 
rapidly  moving  heavy  objects  shall  not  pass  directly  from 
tangents  to  sharp  circular  curves  and  back  again  but 
must  have  their  direction  changed  gradually  over 
transition  curves.  The  transition  curve  has  had  a  long, 
hard  birth,  but  it  is  now  so  easy  to  handle  that  it  may 
be  introduced  into  the  alinement  of  canals,  roads , 
streets,  and  where  the  only  gain  is  in  appearance. 


12  CURVES    OF    CIVIL    ENGINEERING. 

CHAPTER   III. 

LINEAR   MEASUREMENTS. 

7.  All  measurements  should  be  made  with  a  loo-ft. 
steel   tape,  eleven   steel   marking   pins,    a   hand   level, 
and    a    plumb    bob.     The   work    is    done   by   a   head 
tapeman,    rear    tapeman,    stake-marker,    and    axman, 
all  under  the  direction  of  the  head  tapeman,  who  re- 
ceives his  instructions  from  the  transitman  or  assistant 
locating  engineer.     The  0  end  of  the  tape  is  kept  behind 
and  if  plusses  are  to  be  taken  the  stake-marker  drags 
the  tape  ahead  until  the  0  end  is  at  the  last  station. 
Then  the  head  tapeman  reads  the  plus  and  sets  the  point. 

8.  The  rear  tapeman  should  not  be  allowed  to  hold 
his  0  end  of  the  tape  at  any  plus  stakes  or  hubs,  but 
should  always  remain  at  the  last  station  point  until 
the  next  one  ahead  is  set.     This  obviates  the  uncer- 
tainty of  mental  calculations  on  the  part  of  tapemen. 
The    tapemen    do    not    handle    stakes    but    fix    the 
points  in  the  ground  with  the  steel   pins.     The    head 
tapeman   carries   a   range  pole   by  which   the   transit- 
man  puts  him  on  line  when  the  steel  pins  are  not  visible. 
The  stake-marker    drops  a  properly  marked    stake    at 
^each    pin.     The    tapemen     check    these    markings    by 
calling  out  the  numbers  found  on  these  stakes  when 
they   make    the   measurement.     The    axman    follows, 
pulls  up  the  pins  and  drives  the  stakes  in  their  places. 

The  tapemen  should  work  as  close  to  the  ground 
as  possible  and  thus  avoid  sags  in  the  tape,  wind, 
swinging  plumb  bobs,  irregular  tension,  mistaken 
horizon  and  loss  of  time  in  trying  to  locate  points 
of  plumb  bobs.  A  tapeman  who  dislikes  to  stoop 
is  unreliable.  Sometimes,  in  grass  or  brushy  land 
where  the  middle  of  the  tape  is  supported,  it  is 


CURVES    OF    CIVIL    ENGINEERING.  1 3 

best  to  work  from  the  tops  of  stakes  if  there  be  soil 
to  hold  them  firmly.  Even  plumb  bobs  may  some- 
times be  used,  and  all  work  done  in  the  air  to  avoid  cut- 
ting brush.  But  work  in  the  air  and  plumbing  down 
by  either  tapemen  or  transitmen  should  be  avoided 
as  much  as  possible. 

9.  When  on  a   slope  the  rear  tapeman    ''lets  out'" 
the  tape  enough  to  give  horizontal  measurements  and 
when  measuring  on  a  chord   of  a  curve  "pulls  in"  the 
tape  enough  to  give  arc  measurements.     The  amounts 
to  give  or  take  are  shown  in  Table  I  and  Diagram  IV. 
The  rise  or  fall  per  100  is  noted  with  the  hand  level;  one 
tapeman  uses   the   other   as   a  level  rod.      They   soon 
learn  where  the  graduations  are  on  each  man,  though 
he  be  riot  marked.     The  correction  is  inappreciable,  for 
gentle  slopes.     When  the  slope  becomes  more  than  6  per 
hundred  it  becomes  expedient  to  "break  chain,"  using 
not  more  than  50  ft.  of  the  tape  at  one  time,  holding  one 
end  up  to  horizontal  and  plumbing  down.     But  the  previ- 
ous remarks  regarding  plusses  should  be  borne  in  mind 
and  the  rear  end  of  the  tape  kept  at  the  last  station. 

Table  I  is  essentially  the  rear  tapeman 's  table.  No 
one  else  has  any  use  for  it  except  to  know  that  the 
tapemen  use  it  properly. 

10.  The  0  end  of  the  tape  should  extend  0.5  beyond 
the  0  mark  and  the  back  of  the  tape  graduated  for 
slope   and   chord   measuring.     Manufacturers   are   pre- 
pared  to    furnish    them,  giving   the  graduations    back 
from  the   0  mark  for  the  different   slopes   and   ahead 
from  the  0  mark  for  each  degree  of  curve.     The  mov- 
able clamped  index  point  (Lallie's  patent)    should   be 
used  on  the  tape  to  insure  against  the  rear  tapeman's 
forgetfulness  when  on  curves.    If  errors  are  thus  made  by 
his  forgetfulness   it  would   be  no  more  serious  than  if 
made  by  the  engineer's  design,  as  has  been  >done  for 
a  century. 


14  CURVES    OF    CIVIL    ENGINEERING. 

CHAPTER  IV. 

GENERAL  DISCIPLINE. 

11.  Each    man    should    attend   strictly    to    his    own 
duties.     Willingness    to    help    others    is  an  undesirable 
qualification    here.     Few  words    should    be    the    rule, 
and    distracting    talk    prohibited.     When   a   signal   or 
instruction  is  given  it  should  always  receive  a  response 
by  motion  of  arms  or  saying  "All  right"  when  under- 
stood.    Rank  goes  with  salary,  and  no  two  should  have 
the    same"  rank,    so  that    it  will    be  well    understood 
who    controls    movements    when    working    in    detach- 
ments.    They    should    work    as    one    man.     Shouting 
indicates  lack  of  skill. 

12.  Orders    should   be    few    as    possible    and   in    the 
form  of  casual  remarks  or  questions  until  it  appears 
that   an   assistant   does  not  try  to  please.     Then  im- 
perative   orders    become    necessary;    but    they  should 
always   indicate  a  condition  that   cannot  long  endure. 
Keep  watch  for  the  one  man  who  often  disaffects  the 
whole  party.     No  one  man  should  be  allowed,  in  peace, 
to  be  habitually  the  last  at  breakfast  or  at  work.     The 
men   should  be   required  to  bunk  together,    and  their 
baggage  limited  to  simple  necessities. 

Each  assistant  should  feel  sure  of  exact  justice  and 
impartiality;  that  his  comfort  and  welfare  are  given 
careful  consideration.  If  called  upon  to  suffer  hard- 
ship and  exposure,  he  is  not  to  reason  why,  but  feel  that 
it  is  unavoidable.  Results  are  obtained  by  thought  and 
management,  not  by  long  hours  and  labor  alone.  The 
day's  work  should  be  planned  the  evening  before,  and 
started  immediately  after  breakfast  without  any  ques- 
tions or  confusion.  Strict  discipline  is  for  the  good  of 
all,  and  should  be  distasteful  to  none. 


CURVES    OF    CIVIL    ENGINEERING.  1 5 

CHAPTER  V. 

ANGULAR    MEASUREMENTS. 

13.  The    transitman    should    be    relieved    from    the 
discipline     and    management    of   the   party   as   far    as 
possible,  so  that  he  [can  give  ^his   undivided   attention 
to  th'e  mathematics  of  his  work  and  the  adjustments  of 
his  instrument.     A  close  record  should  be  kept  of  the 
azimuth    (or    corrected    courses)    of    all    tangents.      If 
they  are  true  all  his  work  must  be  correct  unless  there 
be  balancing  errors,  which  is  very  improbable.' 

He  should  commence  with  a  true  meridian,  deter- 
mined astronomically.  Then,  as  he  progresses  with 
his  survey,  he  can  check  in  the  same  way  as  often  as 
desired.  The  magnetic  needle  gives  a  rough  check 
and  a  very  useful  one  because  it  is  so  easily  and  often 
applied.  He  should  be  given  every  opportunity  to 
make  complete  ties  with  lines  that  may  have  been  run 
through  the  country  before.  If  there  be  none,  it  may 
be  expedient  to  run  one  (without  chaining),  simply  for 
the  purpose  of  checking  azimuth.  Angles  are  measured 
to  the  nearest  minute,  so  there  is  a  possible  allowable 
error  in  azimuth  equal  to  as  many  half  minutes  as 
there  are  transit  points,  but  such  a  possibility  should 
not  be  considered.  It  is  the  grossest  carelessness  to 
allow  the  azimuth  to  go  unchecked. 

14.  A  complete*  tie  is  illustrated  in  the  sample  transit 
notes,  top  of  page  37.     The  transitman  should  be  given 
vistas  close  to  the  ground  when  setting  transit  points, 
so  that  he  will  not  have  to  rely  upon  the  head  tapeman's 
skill  in  plumbing  down.     The  rear  flag  should  always  be 
on  the  point  last  occupied  by  the  transit.     Otherwise  the 
angles   will   depend   upon   the   work   of    the   tapemen. 
He  should  not  be  required  to  locate  short  spirals  and 


1 6  CURVES    OF    CIVIL    ENGINEERING. 

should  never  be  forced  into  short  sights.  If  transit 
hubs  come  near  together  on  the  line  he  should  be  given 
time  to  fix  sights  upon  distant  objects,  even  if  men 
have  to  be  sent  to  set  such  sights. 

If  chord  measurements  are  used  plusses  for  transit 
points  should  be  avoided  as  much  as  possible.  Plusses  on 
curves  cause  errors,  unless  Table  I  is  used.  Such  errors 
become  magnified  if  the  back  flag  is  not  always  on  the 
point  last  used  by  the  transit.  The  transitman  snould 
be  given  large  discretion  in  everything  that  pertains 
to  the  accuracy  of  the  transit  work.  Detailed'  in- 
structions relieve  him  of  responsibility.  The  chief 
is  apt  to  be  rusty  and  behind  the  times  in  transit  work. 
He  may  advise  but  should  not  crowd  his  ideas  or  assist- 
ants upon  the  transitman.  He  only  cares  to  have  the 
azimuth  check  first,  and  speed  afterwards. 

CHAPTER  VI. 

A  TRUE    MERIDIAN. 

15.  Is  generally  best  obtained  by  means  of  the 
eastern  or  western  elongation  of  Polaris.  This  happens 
twice  a  day  at  times  in  this  country  when  it  is  not  incon- 
venient to  make  the  observation  after  dark  during 
the  seasons  of  field-wrork.  The  elongation  occurs 
when  the  handle  of  the  "  Great  Dipper"  is  due  east 
or  west  of  Polaris.  The  exact  time  can  be  determined 
by  watching  the  star  until  it  apparently  ceases  to 
move  and  changes  its  direction.  The  true  pole  is  then 
between  Polaris  and  the  "Great  Dipper,"  the  angle 
from  Polaris  being  given  in  Table  VIII.  The  cross 
hairs  may  be  illuminated  by  a  common  candle  held 
near  the  object  glass.  The  transit  holds  the  line  until 
daylight,  when  the  result  of  the  observation  is  secured. 
The  vertical  angle  to  Polaris  gives  the  latitude  to  be 
used  in  the  table. 


CURVES    OF    CIVIL    ENGINEERING.  17 


CHAPTER  VII. 

FIELD-WORK. 

1 6.  To  locate   a  curve   on  the   ground    (either  with 
or  without  spirals)  connecting  two  tangents,  the  tangents 
should   be   run   out   to    an   intersection  if  practicable. 
It  is  not  necessary,  but  it  promotes  accuracy  to  work 
from  the  point  of  intersection.     The  P.  I.  is  also  useful 
as   a  permanent  monument   of  the   survey  since  it  is 
generally  out  of  the  way  of  grading  operations  which 
destroy  the  rest  of  the  line. 

17.  The    shortest    line  from  the  P.   I.   to  the  curve 
is  the    External   Secant.      From   the  P.  I.  to  the  ends  of 
the  curve  is  the  Tangent  Distance,  and  between  the  ends 
of  the  curve  is  the  Long  Chord.     These   three  functions 
of  the  curve  are  found  in  Table  IX. 

Even  if  the  P.  I.  is  not  located  on  the  ground  this 
tangent  distance  must  be  shown  in  the  field-notes 
before  the  platting  is  done.  When  the  P.  I.  is  located 
{the  station  stakes  being  set  up  to  the  P.  I.),  the 
transitman  measures  the  whole  angle.  From  Table  IX 
he  takes  the  tangent  distance  corresponding  to  this 
whole  angle.  If  he  is  to  put  in  transition  curves  he 
adds  the  amount  shown  in  the  Table  IX  for  the  spiral 
selected.  This  total  he  divides  by  the  degree  of  curve 
to  be  used  which  gives  his  tangent  distance.  He  then 
instructs  the  head  tapeman  to  measure  this  distance 
along  the  unmeasured  tangent  ahead  and  set  the  P.  T. 
While  the  party  is  doing  this  and  returning,  he  checks 
and  completes  his  calculations  for  the  curve.  He  sub- 
tracts the  tangent  distance  from  the  station  of  the  P.  I., 
which  gives  him  the  station  of  the  P.  C.  or  P.  S.  This 


1 8  CURVES    OF    CIVIL    ENGINEERING. 

he  has  marked  on  a  stake  in  his  presence  and  sends  the 
party  back  to  put  it  in,  pulling  up  the  stakes  as  they  go. 

He  then  moves  his  transit  to  the  P.  C.  or  P.  S.  and 
runs  in  the  curve.  By  adding  the  length  of  the  curve 
to  the  P.  C.  or  P.  S.  he  has  the  station  of  the  P.  T.  first 
set.  This  gives  him  a  perfect  check  on  the  whole  work. 
The  long  chords  are  often  useful  in  passing  obstacles 
that  prevent  measuring  on  the  curve.  If  the  P.  I.  is 
not  used  as  above  described,  a  trial  curve  or  curves 
have  to  be  first  located.  Then,  noting  how  far  to  the 
right  or  left  the  temporary  P.  T.  is  from  the  tangent 
desired,  he  divides  that  distance  by  the  sine  of  the 
whole  angle  of  the  curve  or  curves  run.  This  gives 
him  the  distance  the  P.  S.  or  P.  C.  has  to  be  moved 
along  the  first  tangent  to  bring  the  P.  T.  on  the  second 
tangent  desired.  This  has  been  called  the  "butting 
process."  It  is  practicable  only  where  the  alinement 
is  easy  and  tangents  predominate.  l^ 

1 8.  To  locate  a  long  piece  of  crooked  line,  a  prelim- 
inary line  has  to  be  run,  from  which  complete  to- 
pography notes  are  taken.  An  accurate  contour  map 
is  then  prepared  on  a  scale  of  about  200  ft.  to  one  inch. 
On  this  plat  the  locating  engineer  is  able  to  project  the 
line  for  an  economical  location,  and  determine  in  the 
office  upon  the  position  and  character  of  all  curves.  It 
then  becomes  the  transitman's  simple  duty  to  put  the 
line  on  the  ground.  No  attention  is  then  paid  to  P.  L's, 
as  they  are  usually  out  of  reach  in  elevation  if  not  in 
horizontal  distance  on  such  lines.  But  their  location 
has  to  be  known  by  the  draftsman,  and  the  transitman 
should  make  his  notes  complete  in  this  respect.  When 
the  transit  is  on  a  P.  O.  C.  the  vernier  should  always 
read  0°  when  ranged  to  the  P.  C. 


CURVES    OF    CIVIL    ENGINEERING.  19 

CHAPTER  VIII. 

TRANSITION    CURVES. 

19.  To  be  successful  a  system  for  placing  transition 
curves  in  railway  tracks  must  be  simple,  flexible,  trigo- 
nometrical calculations  avoided,  and  the  work  must  be 
easily  recorded.  The  whole  alinement  of  a  tortuous 
line  should  be  easily  shown  on  a  scale  of  1000  ft.  to  one 
inch.  The  system  should  run  automatically.  It  is  not 
difficult  to  make  a  special  study  of  an  individual  curve 
and,  with  time,  fit  it  out  with  satisfactory  easement 
curves,  but  a  plan  for  keeping  a  record  of  them  and  for 
putting  them  in  by  wholesale  has  not  been  unanimously 
accepted.  Each  railroad  seems  to  have  a  plan  of  its 
own. 

Before  transition  curves  were  used  the  engineer  had 
to  fix  rules  and  give  much  attention  to  the  ' '  run  off ' '  of 
the  superelevation  of  the  outer  rail  at  the  ends  of  curves. 
With  the  advent  of  the  transition  curve  the  homely 
term  fortunately  becomes  obsolete.  Transition  curves 
should  not  be  made  to  fit  the  old  ' '  run  off. ' '  The  lengths 
of  curves  is  unimportant  in  comparison  with  the  con- 
venience in  handling  them. 

The  economical  length  of  a  transition  curve  will  not 
admit  of  mathematical  demonstration.  It  is  a  matter 
of  taste.  The  longer  the  better,  if  it  fits  the  ground; 
as  long  as  the  circular  portion,  is  a  rule  that  favors  tran- 
sit work  and  looks  well.  By  bringing  train  speeds  into 
the  problem  and  fixing  lengths  arbitrarily  the  require- 
ments for  a  successful  system  cannot  be  met . 

20.-  So  the  plan  here  offered  is  to  make  transition 
curves  long  enough,  and  their  lengths  and  other  func- 
tions are  made  to  vary  with  the  main  or  circular  curve ; 


2O  CURVES    OF    CIVIL    ENGINEERING. 

that  is,  inversely  as  the  degree.  This  property  makes 
it  possible  to  tabulate  all  the  dimensions  of  the  com- 
bined curves  and  of  the  individual  spirals  for  i°  curves, 
and  all  the  calculation  that  is  necessary  is  division  by 
the  degree  of  curve. 

Table  IX  not  only  gives  the  usual  functions  for  a 
i°  curve,  but  gives  the  same  functions  for  the  i°  curve, 
combined  with  three  different  types  of  spirals — a  No.  5, 
No.  9,  and  No.  14 — which  is  believed  to  be  all  that  are 
ever  needed  in  railway  practice.  They  have  unreason- 
able dimensions  for  a  i°  curve,  but  they  are  not  expected 
to  be  used  with  a  i°  curve.  When  used  with  sharp 
curves  their  dimensions,  being  divided  by  the  degree 
of  curves,  become  manageable.  Chords  are  100  ft.  long 
for  a  i°  curve,  but  only  10  ft.  long  for  a  10°  curve,  25  ft. 
long  for  a  4°  curve,  etc.  The  No.  5  consists  of  5  chords, 
No.  9  consists  of  9  chords,  No.  14  consists  of  14  chords, 
etc.  They  are  always  located  by  the  same  deflection 
angles  given  on  page  2  2 . 

No.  5  is  suitable  for  the  easy  curvature  of  ''prairie 
roads,"  and  No.  14  is  designed  for  the  sharp  curva- 
ture of  mountain  lines.  These  curves  are  "cubic  par- 
abolas practically,"  with  a  maximum  curvature  of  i° 
when  loo-ft.  chords  are  used,  and  in  all  cases  equal  the 
curvature  of  the  main  or  central  curve.  By  reference 
to  Table  I  it  will  be  seen  that  the  difference  between 
lengths  of  the  chords  and  curves  are  truly  negligible  on 
account  of  the  short  chords.  Since  there  are  no  finite 
arcs,  chord  measurements  are  used  in  this  connection. 
The  No.  5  is  "  Searles  spiral,"  and  the  No.  9  is  used  on 
the  Union  Pacific  Railway  after  Holbrook's  plan. 
They  have  the  deflections  figured  so  that  the  spiral 
can  be  located  from  each  chord-point.  But  it  is  seldom 
convenient  and  never  necessary  to  set  the  transit  on 
the  middle  portion  of  the  spiral.  It  should  not  tax 
the  ingenuity  of  the  transitman  much  to  get  around 


CURVES    OF    CIVIL    ENGINEERING.  21 

all  obstacles  by  means  of  the  dimensions  given  on  page 
23,  though  the  general  deflection  table,  page  25,  will  ac- 
commodate the  most  exacting. 

21.  Selection. — There  is  no  necessity  for  a  great 
variety  of  spirals.  On  a  given  line  of  railroad  the 
variety  of  circular  curves  is  generally  small.  The 
maximum  curves  predominate,  because  the  locating 
engineer  can  generally  save  distance  and  total  curva- 
ture by  using  the  sharpest  curves  allowable.  On  tor- 
tuous mountain  lines  often  75%  of  the  curves  are  of 
maximum  degree.  So  if  a  spiral  be  selected  that  is 
suitable  for  the  maximum  degree  of  curvature,  it  may 
be  generally  considered  suitable  for  the  whole  line. 
It  may  be  unnecessarily  long  for  some  of  the  easier  cur- 
vature, but  the  objection  (if  it  be  an  objection)  is  not 
serious  nor  frequent,  and  the  advantage  of  a  system  is 
great. 

Theoretically  the  longer  the  spiral  the  sharper  the 
central  curve  has  to  be.  But  since  the  longer  spirals 
are  in  connection  with  the  easier  curves  that  objection 
is  void.  The  theory  that  the  length  of  "  run  off  "  should 
vary  as  the  degree  of  curve  is  false,  because  it  rests 
on  the  assumption  that  train  velocity  is  as  high  on 
sharp  as  on  easy  curves.  A  6o-mile  velocity  requires 
twice  the  superelevation  and  length  of  "run  off"  as  a  42- 
mile  velocity,  per  Diagram  III.  So,  when  variation  in 
speed  is  considered,  the  old  "run  off"  is  often  longer 
on  the  easier  curvature. 

If  a  superelevation  of  more  than  7  inches  be  prohibited 
(which  is  customary)  and  if  it  be  undesirable  to  have  an 
excessive  load  on  the  outer  rail,  it  is  undesirable  to  have  a 
velocity 

of  more  than  30  miles  per  hour  on  a  12°         curve. 
'"      "        "     40      "       "      "       "         6°  30'      " 

tt        it  tt       5Q        (t          it         tt          tt  4o 

((  (I  It         /:  ft  (I  It  ((  o  (I 

tl  It  it         o  (i  it  tt  tl  o  tt 


22  CURVES    OF    CIVIL    ENGINEERING. 

This  is  apparent  from  Diagram  III  and  probably  gives 
higher  velocities  than  is  practiced,  or  than  is  safe  on 
ordinary  track. 

A  i°  curve  is  too  easy  anywhere.  A  i-J°  curve  is 
preferable  and  needs  no  easement,  and  may  generally 
be  made  to  take  the  place  of  a  2°. 

The  No.  5  spiral  may  be  used  on  curves  of  less  than 
5°,  the  No.  9  on  curves  from  4°  to  9°,  the  No.  14  on 
curves  from  7°  to  14°,  making  spirals  between  100  and 
200  ft.  long. 

22.  DEFLECTION  ANGLES. 

No.  5  Spiral  is  located  by  the  following  deflection 
angles : 

From  tangent  end  o°  05',  o°  12^',  o°  23^',  o°  37^',  o°  55' 
"       curve       "     o°  25',  o°  47*',  i°  6f,  i°  22*',  i°  35' 

Whole  angle,  2°  30' 

No.  9  Spiral  is  located  by  the  following  deflection 
angles : 

From  tangent  end  3',  7^',  14',  22%',  33',  45i'> 

i°  o',  i°  16*',   i°  35' 
From    curve    end    27',    52^',    i°    16',    i°    37i', 

i°   57',  2°i44',  2°  30',  2°43i/.   2°  55' 

Whole  angle,  4°  30' 

No.  14  Spiral  is  located  by  the  following  deflection 
angles : 

From  tangent  end  2',  5',  9^',  15',  22',  30 J',  40',  51', 
i°  3ff  i°  17',  i°  32',  i°  48^,  2°  6',  2°  25'. 

From  curve  end  28',  55',  i°  20^,  i°  45'.  2°  08',  2°  29f, 
2°  50',  3°  09',  3°  26J',  3°  43',  3°  5*',  4°  n*^4°  35'- 


CURVES    OF    CIVIL    ENGINEERING.  23 

23.  ELEMENTS    OF    INDIVIDUAL    SPIRALS    FOR   A     ic 
CURVE. 

For  elements  of  spirals  for  other  curves  divide  by  the  degree. 


\      / 


A 

500 
900 

B 

C 

D 

E 

F 

G 

H 

/ 

2    30 

4  30 

J 

o     r 

o  55 
i  35 

No.  5  spiral.  .  . 
No.  9  spiral.  .  . 

499-95 
899-74 

8.00 
24.86 

183.3 
316.9 

183.2 
315-9 

66.7 
133.6 

250.0 
449.9 

2.55 
7  .  20 

No.  14  spiral.  . 

1400 

1399-  i 

58.90 

484.0 

480.4 

218.34 

699.1 

16.3 

7  oo 

2  25 

Ordinates  from  tangent  to  each  100  ft.  from  P.  S: 

No.  5  Spiral — 0.14,  0.73,  2.00,  4.40,  8.00. 

No.  9        "    —  .09,  .4,  1.2,  2.6,  4.8,  7.9,  12.2,  17.8,  2  48. 

No.  14  "  —  .06,  .29,  .81,  1.74,  3.20,  5.29,  8.14,  11.86, 
16.57,  22.38,  29.41,  37.78,  47-59.  58-99- 

24.  Space  is  left  in  the  table  for  inserting  the  ele- 
ments of  other  spirals  should  the  three  given  fail  to 
meet  all  requirements.  The  street  railway  people  are 
not  fully  provided  for,  because  their  needs  are  not  under- 
stood and  they  are  wedded  to  a  system  of  ordinates. 
But  with  arc  measurements  they  can  lay  out  their 


24  CURVES    OF    CIVIL    ENGINEERING. 

sharp  curves  in  the  same  way  as  do  other  engineers,  as 
they  have  not  been  able  to  do  before.  So  this  article 
will  be  continued  for  the  benefit  of  those  who  wish  to 
analyze  and  fix  the  tables  for  a  special  use,  to  accord 
with  individual  tastes  and  opinions;  about  "run  off" 
for  instance,  which  has  been  a  live  subject  and  may  be 
kept  alive  indefinitely. 

25.  THE  GENERAL  POLAR  EQUATION 
of  this  curve  is 


(r2     r        i  \ 
_.  _j 1 )    where 
6      4       i2/ 


a  =  first  deflection  angle  from  tangent  (for  first  chord), 

6  =  any  other  deflection  angle  from  tangent  to  chord 
point, 

r  =  number  of  chords  distant  (corresponds  to  radius 
vector) . 

The  second  differential  coefficient  of  this  equation  is 

2d 

— -,  by  which  the  first  column  in  the  table  of  deflections  is 
o 
readily  obtained. 

Let  N —  the  number  of  a  spiral,  which  corresponds 
to  the  number  of  loo-ft.  chords  that  are  necessary  to 
make  a  maximum  curvature  of  i°  at  the  end,  then 

N° 
Whole  angle  of  spiral  —  - — 


a 
30 


a  — 


N  +  i  , 

5' 
sin  (2^' +5)'. 


CURVES    OF    CIVIL    ENGINEERING. 


Spirals  increase  tangent  distances  G  +11  tan  -J  whole 
angle. 

Spirals  increase  external  secants  -  •. 

cos  -j  whole  angle 

Thus  figures  relating  to  spirals  are  inserted  in  Table  IX 

26.  GENERAL  DEFLECTION   TABLE. 
(A  table  of  coefficients.) 


Different  Positions  of  Transit. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

10 

p.  s. 

0 

i 

3i 

7i 

12* 

19 

26f 

36 

46* 

58* 

74 

i 

i 

O 

2 

sri 

io| 

16* 

24 

32* 

43 

S4* 

67* 

2 

*i 

2 

O 

3 

7* 

13* 

20* 

29 

38| 

So 

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This  is  an  actual  table  for  a  No.  29  spiral  where  a  =  i. 
To  use  the  table  for  any  spiral  multiply  all  the  tabular 

quantities  by  a ;  or    3°    . 

That  is,  for  a  No.  5  spiral  multiply  the  tabular  quantities" 
by  5',  for  a  No.  9  spiral  by  3',  for  a  No.  14  by  2',  etc. 

The  different  positions  of  transit  are  shown  thus  © , 
from  which  read  up  for  deflections  toward  the  tangent 
and  down  for  deflections  toward  the  curve.  This 
table  is  easily  extended  by  use  of  the  constant  differences 
which  exist  in  the  diagonal  rows  of  figures  parallel  with  ©  . 
These  constant  differences  equal  the  number  of  rows 
from  © . 

Other  interesting  properties  of  this  curve  are  ably- 
presented  by  Torrance,  Vial  and  Fulton  in  Vol.  VII. 
No.  2  Journal  of  the  Western  Society  of  Engineers.  ' 


26  CURVES    OF    CIVIL    ENGINEERING. 

CHAPTER  IX. 

PLATTING. 

27.  The  draftsman  should  plat  the  line  under  the 
direction  of  the  transitman.  The  transitman  should 
not  lay  his  more  or  less  perfect  notes  upon  the  drafts- 
man's table  and  feel  that  he  is  through  with  them. 
The  draftsman  should  not  handle  the  transit  notes, 
but  they  should  be  read  to  him,  leaving  him  free  to 
watch  his  points  and  handle  his  tools,  and  allowing  the 
transitman  to  keep  his  books  until  full,  and  making 
the  notes  continuous.  With  proper  system  a  day's 
work  can  thus  be  platted  in  half  an  hour.  The  chief  of 
the  party  has  much  to  do  with  the  draftsman  directly, 
and  he  should  be  careful  that  the  draftsman  does  not 
get  a  wrong  idea  of  his  position,  that  he  may  not  think 
that  instrument-men  are  under  his  care,  or  that  he  is 
independent  of  them.  When  lines  do  not  come  together 
on  paper  as  they  do  on  the  ground  it  is  the  transitman1  s 
duty  to  find  the  error  with  the  draftsman's  assistance, 
and  himself  explain  to  the  chief  of  the  party.  Much 
confusion  is  caused  by  separating  the  field-work  and 
drafting  both  in  camp  and  on  maintenance  work. 
Draftsmen  and  fieldmen  too  often  make  trouble  for 
each  other  and  the  company. 

The  transitman  should  prepare  a  schedule  from  his 
notes  showing  the  courses  and  distances  of  all  tan- 
gents from  P.  I.  to  P.  I.,  and  on  long  chords  of  com- 
pound curves.  From  this  sheet  the  draftsman  can 
plat  alone  if  the  transitman  is  pressed  for  time.  He 
first  draws  a  light,  straight  line  through  the  center 
of  the  roll  of  detail  paper.  A  course  is  given  to  this  line 
corresponding  to  the  general  direction  of  the  survey. 


CURVES    OF    CIVIL    ENGINEERING.  2J 

From  this  line  all  courses  are  taken  by  means  of  the 
protractor  and  transferred  with  the  triangle  and  straight- 
edge or  parallel  rule  to  any  part  of  the  map  desired. 

28.  When  the  lines  are  all  platted  as  described  in  the 
schedule  then  the  tangents  are  scaled  off  from  the  P.  I.'s, 
and   the   centers   of   curves  located.     When  transition 
curves  are  used  it  is  more  convenient  to  draw  the  secant 
lines  and  to  locate  the  centers  of  curves  on  them  by 
scaling  first  from  the  P.  I.  the  external  secant  to  the 
curve,   then  the  radius  to  the  center  required.     Then 
draw  the  circular  curves  with  the  compass  and  the  tran- 
sition curves  with  the  spiral  rule.     Write  the  station 
and  plus  for  each  P.  C.,  P.  S.,  and  P.  T.  in  the  radial 
lines,   and  the  angles  and  degrees  between  the  radial 
lines.     Then,  with  the  spring  ^dividers  and  scale,  locate 
every  even  tenth  station  from  0.     This  is  done  only  for 
a  check,  and  need  not  be  inked  in  if  there  be  crowding 
of  figures.     If  the  work    is   done  in  the  order  named 
any   possible   error  in   platting   or   calculating   will   be 
detected.     The   transitman  may  then  leave   the  work 
to  the   draftsman   and  topographer.     The  line   should 
be  inked  in  with  vermilion  water  color,  but  the  station 
numbers  should  be  left  in  pencil  so  their  position  may 
be  shifted  if  they  be  found  to  interfere  with  topography 
notes.     Letters  and  figures  of  all  kinds  should  be  made 
to  read  from  the  southerly  side  of  the  plat.     A  simple 
record  of  the  spiral  angle  gives  complete  information 
regarding    the    spiral.     The    number    might    be    given, 
but  that  would  be  twice  the  whole  angle  of  the  spiral 
in  degrees. 

29.  As  has  been  intimated,  when  compound  curves 
with  spirals  are  encountered,  the  long  chords  of  curves 
are  platted  instead  of  the  tangents.     Because  tables  can- 
not then  be  used  in  determining  tangent  distances,  so 
it  becomes  a  special  problem  quite  complicated.     First 
the  chord  of  the  spiral  is  platted,  then  of  the  first  cir- 


28  CURVES    OF    CIVIL    ENGINEERING. 

cular  curve,  then  of  the  next  circular  curve  or  curves, 
and  finally  of  the  last  transition  curve. 

This  method  of  platting  also  has  to  be  used  if  the 

locating  engineer  thinks  there  should  be  different  kinds 
of  spirals  at  the  two  ends  of  a  simple  curve,  or  if  a  spiral 
has  to  be  located  between  two  parts  of  a  compound 
curve.  These  complications  ought  to  be  avoided,  and 
can  be  with  a  little  ingenuity.  Compound  curves  can- 
not be  avoided,  but  connecting  curves  of  great  difference 
in  degree,  necessitating  a  spiral,  can  be  avoided  by  put- 
ting in  one  or  two  additional  P.  C.  C.'s.  For  those 

.  who  disagree  and  have  time  for  the  confusing  calcula- 
tions use  the  No.  9  or  No.  14  spirals.  The  chord-lengths 
are  made  to  correspond  to  the  sharper  curve;  that  is, 
100  divided  by  the  degree.  A  part  is  taken  off  from 
the  tangent  end  of  the  spiral  proportional  to  the  smaller 
degree.  That  is  if  a  3°  is  connected  with  a  9°  curve 
three  chords  will  be  taken  off  the  tangent  end  of  the 
No.  9  spiral.  Here  the  complete  deflection  table  is 
necessary  to  turn  deflections  from  the  third-chord 
point.  The  field-work  is  simple.  The  computations 
for  platting  are  by  main  strength  and  awkwardness 
of  latitudes  and  departures. 

Convenience  of  platting  should  be  considered  in  all 
field-work.  A  large  percentage  of  surveys  are  wasted 
because  they  cannot  be  easily,  intelligently  platted. 


CHAPTER  X. 

TURNOUT    CURVES. 

30.  An  ordinary  standard-gage  turnout  from  tan- 
gent first  makes  an  angle  of  about  2°,  then  runs  straight 
1 6  ft.  along  the  split  rail,  then  there  should  be  a  true 
circular  curve  to  the  wing  of  the  frog,  then  a  tangent 


CURVES    OF    CIVIL    ENGINEERING.  29 

for  the  full  length  of  the  frog,  about  13  ft.,  and  on  to 
clearance  or  to  the  next  frog  if  it  be  a  cross-over.  The 
lines  of  a  frog  are  straight  and  rigid.  This  alinement 
is  further  complicated  by  widening  the  gage  on  the 
curve  and  at  the  point  of  frog. 

So  it  is  impractical  to  do  this  work  with  the  transit. 
The  lead  rail  is  located  by  ordinates  from  the  main- 
line rail.  The  whole  angle  of  the  lead-rail  curve  equals 
the  angle  of  the  frog,  minus  the  angle  of  the  split  rail. 
The  length  of  the  curve  depends  upon  the  dimensions 
of  the  frog,  the  split  rail,  and  the  pattern  of  rail.  It  is 
quite  simple  to  design  a  standard  plan  when  all  these 
data  are  known.  There  would  be  only  two  or  three 
such  plans  for  a  large  railroad  system.  Crotch'  frogs 
are  generally  prohibited.  If  used  they  break  up  the 
lead  into  two  curves  with  a  tangent  over  the  frog.  Turn- 
outs from  curves  are  avoided  if  possible,  but  if  unavoid- 
able the  standard  plan  may  be  used  the  same  as  if  on 
tangent.  The  transit  work  should  begin  at  the  P.  I. 
where  the  tangent  passing  the  frog  intersects  the  center 
of  main  line.  There  turn  the  frog  angle,  run  past  the 
frog,  after  which  curves  may  be  started  as  desired. 
Tables  cannot  be  used  except  for  approximate  work. 
There  are  no  practicable  special  problems.  They  are 
all  spoiled  by  the  frog  tangents,  if  not  by  the  split  rails. 

31.  The  transitman  should  not  be  intimidated  by 
the  mass  of  figures  and  formulas  that  are  sometimes 
used  to  illuminate  this  subject.  The  problems  are  all 
quite  simple  and  not  different  from  those  encountered 
-elsewhere  in  curve  location.  For  example,  to  stake 
out  the  grading  the  turnout  curves  may  be  considered 
,as  having  a  whole  angle  equal  to  the  frog  angle,  and  an 
external  secant  equal  to  J  gage.  With  these  data  enter 
Table  IX  and  find  the  degree  and  length  of  curve  as 
•close  to  the  truth  as  can  be  obtained  by  any  special 
turnout  tables,  or  formulas,  but  only  close  enough  to  use 


30  CURVES    OF    CIVIL    ENGINEERING. 

•  in  grading.  Yard  maps  cannot  be  made  until  after 
track  laying,  since  rail- joints  fix  the  location  of  switches. 
Ill-founded  theories]  and  formulas  regarding  turn- 
outs has  done  much  to  take  track-work  out  of  the  hands 
of  engineers.  It  has  been  one  cause  of  the  vexatious 
reference  to  the  difference  between  theory  and  practice . 
There  would  be  no  difference  if  the  engineer  was  left 
to  figure  out  his  own  special  problems  and  understand 
the  foundation  of  all  theories.  Rules  and  tables  are 
dangerous  in  the  hands  of  those  unable  to  reproduce 
them  independently. 


CHAPTER  XI. 

VERTICAL    CURVES. 

32,  Vertical  curves  are  laid  out  by  vertical  ordinates 
from  a  horizontal  line  with  the  level.  They  are  usually 
very  flat  on  railroads  in  comparison  with  horizontal 
curves.  The  sharpest  vertical  curve  allowed  on  at  least 
two  prominent  railroad  main  lines  has  a  radius  of 
1 14, ooo  ft.  This  accounts  for  the  simple  formulas  and 
definitions  in  this  connection. 

The  curvature  of  vertical  curves  is  designated  by  num  - 
bers  in  place  of  degrees.  The  Number  of  a  Curve  is 
its  change  in  its  rate  of  grade,  expressed  in  hundredths 
of  a  foot,  per  hundred  feet.  For  example,  a  No.  7  curve 
changes  its  rate  of  grade  .07  each  successive  100  ft. 
This  .07  corresponds  to  the  chord  deflection  in  hori- 
zontal curves.  If  the  grade  be  assumed  to  run  on  the 
loo-ft.  chords  of  the  curve  the  rate  of  grade  on  the  first 
chord  from  tangent  will  be  changed  in  hundredths, 
only  one  half  the  number  of  the  curve,  for  the  same  rea- 
son that  the  tangent  deflections  are  half  of  chord  deflec- 
tions. But  for  all  succeeding  chords  the  rate  of  grade 


CURVES    OF    CIVIL    ENGINEERING.  3! 

changes  as  many  hundredths  as  the  number  of  the 
curve. 

The  Whole  Angle  is  designated  by  tenths  of  a  foot 
per  hundred  in  place  of  degrees.  For  example,  if  a  —  0.8 
grade  intersects  a  +0.9  grade  the  angle  formed  will  be 
17  tenths. 

If  A  =  whole  angle,  N  =  number,  5  =  external  secant 
in  tenths,  and  L  =  length  of  curve  in  stations,  then 


and  S=±AL. 


Approximate   radius   in   inches    on    Plate   A   profile 

T25 
paper  =  -^-. 

wi 

By  these  formulas  and  Table  VII  the  engineer  plats 
the  grade  line  on  the  profile  and  calculates  the  eleva- 
tion of  grade  for  each  station  before  going  into  the 
field.  The  P.  I.'s  should  be  made  at  even  stations, 
and  the  ends  of  curve  can  usually  be  made  at  stations 
without  causing  faulty  grades.  This  greatly  simplifies 
the  calculations  which  have  to  be  tried  and  repeated 
until  the  grade  line  is  satisfactory.  Straight  grades 
can  usually  be  expressed  with  one  decimal  place,  and 
never  more  than  two  should  be  used,  except  when 
compensating  for  curvature  on  maximum  grades. 

33.  When  laying  a  maximum  grade,  or  where  uniform 
resistance  is  desired,  the  rate  of  grade  should  be  made 
less  on  curves  than  on  tangents  by  the  amounts  shown 
in  column  8,  Table  VI.  In  the  lower  portions  of  such  a 
grade,  where  high  velocities  are  admissible,  and  can  be 
had,  compensation  for  curvature  is  unnecessary.  But 
where  the  velocity  may  fall  to  eight  miles  per  hour  curve 
resistance  should  be  fully  compensated  in  the  grade,  so 
that  the  resistance  and  velocity  will  be  uniform.  If 
the  velocity  drops  so  the  centrifugal  force  is  lost  the 


32  CURVES    OF    CIVIL    ENGINEERING. 

train  "stalls."  A  maximum  grade  in  a  district  is  not 
necessarily  one  of  maximum  ratio,  but  one  which  pro- 
duces a  maximum  tractive  power,  where  curves  and 
velocity  are  considered.  It  is  desired  to  avoid  enter- 
ing the  field  so  ably  covered  by  the  late  A.  M.  Welling- 
ton in  his  Economic  Theory  of  Railway  Location;  but 
that  statement  is  necessary  to  support  the  new  principle 
advocated,  viz. : 

34.  That  more  easy  vertical  curves  should  be  intro- 
duced into  railway  grades  to  supersede  long,  straight 
grades  which  cause  sharp  summits  and  sags.  Long,  easy, 
vertical  curves,  reversing  in  the  middle  of  a  hill,  can  be 
used  with  great  advantage,  especially  if  the  rate  of  grade 
at  the  P.  R.  C.  be  not  held  down  to  the  same  rate  allow- 
able on  a  straight  grade  several  miles  long.  ''  Mo- 
ment um  "  grades  can  be  used  nowhere  to  a  better  advan- 
tage than  at  a  P.  R.  C.  of  vertical  curves.  Curves  are 
more  difficult  to  plot  upon  the  profile,  but  they  are  well 
worth  the  trouble.  A  large  percentage  of  straight 
grade  generally  causes  sharper  curves,  which  are  very 
objectionable  in  a  grade  line.  This  is  for  the  same 
reason  that  a  sharper  curvature  gives  larger  percentage 
of  tangent  in  horizontal  alinement.  Establishing  grade 
is  a  capital  service,  where  skill,  time,  and  care  are  well 
spent  in  projecting  curves. 


TABLES 

AND 

DIAGRAMS. 


33 


TABLE  I. 


DIFFERENCES   IN    LENGTHS    OF   ARCS    AND 
CORRESPONDING   CHORDS. 


"o  6 

V   > 

F 

Lengths   of  Arc. 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

1° 

.OOI 

.OOI 

.OOI 

.OOI 

2° 

.005 

.004 

.003 

.002 

.001 

3° 

.Oil 

.009 

.006 

.OO4 

.002 

.001 

4° 

.020 

.015 

.010 

.007 

.004 

.002 

.001 

5° 

.031 

.023 

.016 

.Oil 

.007 

.004 

.002 

6° 

.045 

•  033 

.023 

.015 

.010 

.006 

.003 

.001 

7° 

.062 

•  045 

.031 

.021 

.013 

.008 

.004 

.002 

8° 

.081 

•°59 

.041 

.028 

.017 

.010 

.005 

.002 

9° 

.103 

•°75 

.053 

.036 

.022 

.013 

.006 

.003 

10° 

.127 

•093 

.065 

.044 

.027 

.016 

.008 

.003 

.OOI 

11° 

.154 

•113 

.079 

.053 

•033 

.019 

.010 

.004 

.OOI 

12° 

.184 

.134 

.094 

.063 

.040 

.022 

.012 

.005 

.OOI 

I3o 

.216 

.158 

.  no 

.074 

.046 

.027 

.014 

.006 

.002 

14° 

.250 

.183 

.127 

.086 

•°54 

•  032 

.016 

.007 

.OO2 

15° 

.286 

.  209 

.  146 

.099 

.062 

.036 

.018 

.008 

.OO2 

16° 

.324 

.236 

.166 

.  112 

.070 

.041 

.021 

.009 

.003 

I7o 

.365 

.266 

.187 

.  126 

.079 

.  046 

.023 

.010 

.003 

18° 

.409 

.298 

.209 

.141 

.088 

•051 

.026 

.Oil 

.003 

19° 

•  456 

•333 

.233 

.157 

.098 

.057 

.029 

.012 

.OO4 

20° 

•507 

•370 

.259 

.174 

.  109 

.064 

.032 

.014 

.004 

25o 

.792 

.580 

.407 

•*73 

.172 

•099 

.051 

.021 

.OO6 

30° 

1.136 

.830 

.580 

•39° 

•245 

.142 

.072 

.031 

.OO9 

.OOI 

40° 

2.018 

i  .480 

1  .036 

•695 

•435 

.252 

.130 

.054 

.Ol6 

.002 

50° 

3**44 

2  .  270 

1  .60 

i  .086 

.686 

•393 

.  200 

.085 

.025 

.003 

60° 

4.507 

3.300 

2.31 

i-55o 

.970 

.564 

,290 

.121 

.036 

004 

100° 

12  .2l8 

180° 

36.3 

This  table^was  calculated  by  the  following  formula. 

1  1459.1  6  X  sin  jf  degree 

ov  ,  — 
degree; 

34 


erence  =  100 


II.  SAMPLE  TRANSIT  NOTES. 


*7an.  5  '03 
STATION 

DEF. 

C.C. 

M.C. 

+  44  P.T. 
9 

2°55' 

2°22' 

N.  20°20'  E. 

8 

0°8' 

Whole 
Cir.  Cur 

Angle 
'e     16°12' 

4-  94  P.S. 

7 

8°06' 

5°17' 

3 

Spirals 
Tots 

9°00' 

1      25°12/ 

6 

2°17' 

$  £ 

is. 

Tang. 

288.7 

4-  24  P.O. 
5 

1°35' 
1°09' 

0°      A 

CO      ."ti 

1 

P.I.  286  H 

-  62.7  not  set 

^ 

4 

o°e' 

iM 

f-  74  P.S. 
3 

' 

2 

1 

280 

9        P.O.T. 

N.  SS^SO'V 

8 

N.  45032'^E. 

7    -f-  00  P.T. 

2°55' 

6 

1°57' 

5- 

0°27' 

^2 

J    S 

Whole 
Cir.  Cur 

Angle 
re      5°00' 

+  75  P.S. 
4 

2°30' 

•Too' 

0   "n 
E   M 

h  ^ 

Spirals 
Tot 

9°00' 

il      14°00/ 

+  50  P.O. 
3 

1035' 

l^' 

s* 

°*  * 

Tang. 

288.6 

^%/ 
^<J 

2  . 

0°14' 

E 

P.I.  274- 

-13.6 

\*%P 

^.yXc^P 

4-  25  P.S. 

270 

N.  59°32'E. 

9 

N.  49°30': 

-j-  40  P.T. 
8 

1°3Q' 
1°18' 

'V* 

7 

48' 

<B 
> 
M 

Whole  ^ 
Tang. 

ngle  3°00' 
150.1 

6 

18' 

0 

P.I.  266 

-90.1 

+•  40  P.C. 
265 

N.  56°32'E. 

Jan.  5  ''OS 
STATION 

DEF 

C.C. 

M.C. 

4 

N.  89°29'  E. 

-f-  40  P.T. 
3 

4°35' 

3°09' 

2  4-  00  P.S. 

18°00' 

Whole 
Cir.  Curv 

Angle 
e     36°00' 

1 

13°00' 

CO 

ti  1 

Spirals 
Tot£ 

uW 

1     50°  OQ' 

no 

8°00' 

g  & 

%  3 

9 

3°00' 

O  ^ 
o    g 

Secant 

61.00 

4-  40  P.O. 

8 

2°25' 

oV 

P 

Tang. 
P.I.      31C 

337.8 
-f-  37.8 

€& 

7+0  P.S. 

S^P 

^Y<$> 

6 

1ST.  39°29'  E. 

5-h  00  P.T. 

1°35' 

N.  29°30'"E. 

4  4-  00  P.S. 

8°ioH'  ; 

Whole 
Cir.  Cur\ 

Angle 
e     16°2l' 

3 

5°40^' 

CO 

^  1 

Spirals 
Tot* 

5°00' 

y&z 

1     21°2l' 

4-  50  P.O.C. 

2 

4°25^' 
3°10^' 

o     P* 

fc  ^ 

g      0 

Ex.  Sec. 

20.7' 

HifX 

$?$* 

1 

0°40J4' 

U  * 

°^    H 
'{>. 

Tang. 

266.1 

I 

4-  73  P.O. 

00 

0°55' 
0°05' 

4  73P.R.C. 

9 

2°55/ 
l°4l' 

4-  48  P.S. 

8 

15°45' 
13C50' 

Cir.  Cur\ 

e     31°  30' 

7 

9°50' 

«1 

Spirals 
Toti 

9°00' 
1     40°30/ 

6 

5050' 

§  £ 

2  o 

P.I.    296- 

1-50  set 

5 

1°50' 

°oo    3 
*fc 

Tang. 

320.8 

4& 

^LJjfJPliu 

4-  54.2  P.O. 
4- 

1°35' 
0032' 

<^b/c\* 

%Y^' 

4-  29.2  P.S.' 
3 

2 

1 

N.  20°20'  E. 

DO 

• 

N.  10°15'  E. 

•4-  44  P.T. 

II.  SAMPLE  TRANSIT  NOTES. 


Lin 


B 

264+ 


v/ 

m. 


*£ 


* 


266  +  30 


25  +20, 


248+12 


jar 

Ston 


38 


III.  SUPERELEVATION  OF  OUTER  RAIL  ON  CURVES. 

STANDARD  GAUGE  TRACK. 
IN  INCHES  =  DEGREE  x  VELOCITY2  (IN  MI.^PER  H.)  x  .00065. 

Superelevation  for  other  gauges  are  proportional  to  gauge. 

INCHES    SUPERELEVATION 
01  2345  67 


39 


IV.    DIAGRAM  OF  CORRECTIONS  FOR  MEASURING. 


ON  SLOPES  ADD 

TENTHS  OF  FOOT' 

1 


.4    A 


ON  CHORDS  OF  ARCS  DEDUCT 

TENTHS  OF  FOOT. 

2  12 

18° 


40 


TABLE  V. 

AN    EMERGENCY   TABLE    FOR  DETERMINING 
NATURAL   FUNCTIONS  OF  ANGLES. 

Sine  obtained  directly  from  the  table; 
Cosine  =sine  of  complement  of  angle; 

™  sine 

Tangent  =  —         or 
cos 

_  tangent  distance  for  double  the  angle  in  Table  IX. 
lL5729-6   I 

^    j.  COS  I 

Cotangent  =  — —  = ; 

sine     tang 

becant  = ;— ; 


Cosecant  =  — — ; 
sine 

Versine  =  i  —  cosine ; 
Coversine  =  i  —  sine. 


EXAMPLE. 


Required  sine  of 

From  table  36°  09^'      =.59000 


Correct  to  four  decimal  places. 
41 


TABLE  V. 
NATURAL   SINES. 


Angle, 
o    / 

Sine. 

Dif.  i'. 

Angle. 

0     / 

Sine. 

Dif.  i'. 

Angle. 

0      f 

Sine. 

Dif.  i'. 

o  344 

.01 

.00029 

20  29 

•35 

.00027 

42  51 

.68 

.00021 

i  09 

.02 

*  ' 

21  06 

.36 

<  i 

43  38 

.69 

.OOO2I 

I  43 

•03 

t  < 

21  43 

•37 

1  ' 

44  26 

.70 

.OOO2I 

2  174 

.04 

« 

22  20 

•38 

" 

45  14 

•71 

.00621 

2  52 

•05 

'  * 

22  57 

•39 

'  ' 

46  034 

•72 

.OOO2O 

3  26J 

.06 

'  * 

23  35 

.40 

'  ' 

46  53 

•73 

.OOO2O 

4  01 

.07 

<  c 

24  124 

.41 

'  ' 

47  44 

•74 

.OOO2O 

4  354 

.08 

(i 

24  50 

.42 

.00026 

48  354 

•75 

.00019 

5  10 

.09 

" 

25  28 

•43 

'  ' 

49  28 

.76 

.OOOI9 

5  44i 

.10 

*  * 

26  06 

.44 

'  ' 

50  21 

•77 

.OOOI9 

6  19 

.11 

« 

26  444 

•45 

*  ' 

51  16 

.78 

.OOOlS 

6  534 

.  12 

« 

27  23 

.46 

« 

52  ii 

•79 

.00018 

7  28 

•13 

** 

28  02 

•47 

« 

53  08 

.80 

.OOOI7 

8  03 

.14 

*  * 

28  41 

.48 

.00025 

54  06 

.81 

.OOOI7 

8  374 

•15 

'  ' 

29  2©4 

.49 

'  ' 

55  05 

.82 

.OOOI7 

9  124 

.16 

'* 

30  oo 

•5° 

'* 

56  06 

•83 

.00016 

9  47 

•17 

*  * 

30  40 

-5i 

" 

57  084 

.84 

.OOOl6 

10  22 

.18 

*  * 

31  20 

•52 

11 

58  13 

•85 

.OOOI5 

10  57 

.19 

*  * 

32  oo 

•53 

'  ' 

59  19 

.86 

.OOOI5 

II  32 

.  2O 

'  * 

32  41 

•54 

.00024 

60  274 

.87 

.00014 

12  074 

.21 

" 

33  22 

•55 

" 

61  39 

.88 

.OOOI4 

12  434 

.22 

.00028 

34  034 

•56 

11 

62  52 

.89 

.OOOI3 

13  18 

•23 

*  * 

34  45 

•57 

M 

64  10 

.90 

.OOOI3 

13  53 

.24 

** 

35  27 

•58 

11 

65  30 

.91 

.00012 

14  29 

•25 

11 

36  094 

•59 

.00023 

66  56 

.92 

.OOOI  I 

15  °4 

.26 

" 

36  52 

.60 

" 

68  26 

•93 

.OOOII 

15  40 

.27 

*  * 

37  354 

.61 

" 

70  03 

.94 

.OOOIO 

16  154 

.28 

" 

38  IQ 

.62 

11 

7i  48 

•95 

.00009 

16  514 

.29 

« 

39  03 

•63 

*' 

73  45 

.96 

.00008 

17  274 

•30 

*  * 

39  474 

.64 

.OOO22 

75  56 

•97 

.00007 

18  034 

•31 

*  * 

40  324 

•65 

'  ' 

78  3i4 

.98 

.00006 

18  40 

•32 

" 

41  18 

.66 

M 

81  54 

•99 

.00004 

19  16 

•33 

.00027 

42  04 

.67 

" 

90  oo 

i  .00 

.00000 

19  524 

•34 

42 


TABLE  VI. 
CURVE    CHARACTERISTICS. 


For  a  i  oo  -Foot  Arc. 

Grade 

D^> 

Diff.  in 

of 

Track 

De- 
gree. 

Radius. 

Middle 
Ordi- 
nate. 

Quar- 
ter Or- 

dinate. 

Tang. 
Dis- 
tance. 

Chord. 

of  Rails 
per    100 
Feet. 

alent 
Re- 
sist- 
ance. 

Gauge. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

0 

/       // 

i 

11459.16 

.109 

.082 

.44 

100.000 

.0410 

0.015 

4     8* 

I 

5729-58 

.218 

.163 

.87 

99.999 

.0820 

0.03 

4     8£ 

if 

3819.72 

.327 

.245 

I-3I 

99-997 

.1230 

0.045 

4     8* 

2 

2864.79 

-436 

.328 

1.74 

99-995 

.  1640 

0.06 

4     8i 

2* 

2291  .84 

-545 

.408 

2.18 

99.992 

.  2050 

0.075 

4     8f 

3 

1909.86 

.655 

.490 

2.62 

99.989 

.  2461 

0.09 

4     8|- 

si 

1637  .02 

.764 

-573 

3-05 

99.985 

.2871 

0.  10 

4     8* 

4 

I  43  2.  "40 

-873 

-655 

3-49 

99  .  980 

.3281 

0.  II 

4     83- 

S 

1145-92 

i  .  090 

.82 

4.36 

99.969 

.4101 

0.14 

4     8* 

6 

954-93 

1.309 

.98 

5.23 

99-955 

.4922 

o.  16 

4     8* 

7 

818.51 

1-53 

-15 

6.  10 

99.938 

•  5742 

0.18 

4     8* 

8 

7l6.  20 

i-75 

-38 

7.0 

99.919 

.6562 

0.  20 

4     8f 

9 

636.62 

i  .96 

•47 

7-9 

99.897 

.7383 

0.  22 

4      9 

10 

572.96 

2.18 

.63 

8.7 

99.873 

.8203 

o.  24 

4      9 

II 

520.87 

2.40 

.80 

9.6 

99.846 

.9024 

o.  26 

4      9 

12 

477.46 

2.62 

-97 

10.4 

99.  816 

.9844 

0.28 

4      9 

13 

440-74 

2.84 

•13 

ii.3 

99.784 

i  .0664 

0.30 

4      9i 

14 

409.25 

3-05 

.29 

12.2 

99-750 

1.1484 

0.32 

4      9* 

15 

381.98 

3-27 

.46 

13.0 

99.714 

i  .  2300 

0.34 

4      9* 

16 

358.10 

3-49 

.62 

13-9 

99.676 

1.3124 

0.36 

4      9± 

18 

3I8.3I 

3-92 

2.94 

15-6 

99-591 

1.4766 

0.38 

4      9i 

20 

286.48 

4-36 

3.28 

17.4 

99-493 

i  .6406 

0.40 

4     9* 

25 

229.18 

5-43 

99.  208 

30 

190.99 

6.51 

98.864 

35 

163.70 

7.6i 

98.464 

40 

143.24 

8.64 

97.98i 

50 

114-59 

10.75 

96.856 

60 

95-49 

12.80 

95-493 

180 

31-83 

31-83 

63.66 

Columns  7,  8,  and  9  refer  to  standard  gauge  railroad  track. 

Column  5  is  the  distance  of  curve  from  tangent  100  feet  from  point  of 
tangency  and  equals  half  the  chord  deflection. 

To  find  any  radius  not  here  shown,  divide  5729.58  by  the  degree  of 
curvature. 

To  find  radius  of  an  old-style  curve  located  witu  IOO-IT;.  cnords,  add 
07  X  degree  of  curve  to  tabular  quantity. 

4? 


TABLE  VII. 

ORDINATES    FROM    TANGENT   TO    VERTICAL 
CURVES  FOR  EACH  100  FEET  FROM  P.  C. 


6 

fc 

100 

200 

300 

400 

500 

600 

700 

800 

900 

1000 

IIOO 

.60 

1200 

1300 

Q  . 

1400 
nc 

2 

.OI 

.04 

.09 

.16 

•25 

.36 

•49 

.64 

.81 

.50 

I.OO 

1.  21 

•7  2 

1.44 

.04 
I  .69 

•9° 
1  .96 

2* 

.01 

•OS 

.11 

.20 

.31 

•45 

.61 

.80 

1.  01 

1.25 

i-5i 

1.  80 

2.  II 

2-45 

3 

.01 

.06 

.14 

•24 

.38 

•54 

•74 

.96 

1.  21 

1.50 

1.81 

2.16 

2.53 

2.94 

3* 

.02 

.07 

.16 

.28 

•44 

•63 

.86 

.12 

1.42 

1.75 

2.12 

2.52 

2.96 

7    78 

3-44 

4* 

.02 

.09 

.20 

.36 

.56 

.81 

.10 

•44 

1.82 

2.25 

2.72 

3.24 

6  -o° 
3.80 

3.92 

4-41 

5 

•03 

.10 

.22 

.40 

.62 

.90 

.22 

.60 

2.02 

2.50 

3-02 

3.60 

4.22 

4-89 

Si 

•03 

.11 

•25 

.44 

.69 

•  99 

•  35 

.76 

2.23 

2.75 

3-33 

3.96 

4-65 

5-39 

6 

•03 

.12 

.27 

.48 

•  75 

.08 

•47 

.92 

2.43 

3-oo 

3.63 

4.32 

5-07 

5-38 

6* 

•03 

•  13 

.29 

•  52 

.81 

•17 

•  59 

.08 

2.63 

3-25 

3-93 

4.68 

5-49 

6-37 

7 

.04 

.14 

•31 

.56 

.87 

.26 

•  71 

.24 

2.83 

3-50 

4-24 

5-05 

5-91 

6.86 

7* 

.04 

•  IS 

•34 

.60 

•93 

•  35 

.84 

.40 

3-04 

3-75 

4-54 

5-40 

6.34 

7-35 

8 

.04 

.16 

.36 

.64 

.00 

•44 

.96 

•  56 

3-24 

4.00 

4.84 

5.76 

6.76 

7.84 

8* 

.04 

.17 

-38 

.68 

.06 

•53 

.08 

•  72 

3-44 

4-25 

5.14 

6.12 

7.18 

8.33 

9 

•05 

.18 

.40 

.72 

.12 

.62 

.20 

.88 

3-64 

4-50 

5-44 

6.48 

7.60 

8.82 

9* 

•OS 

.19 

•43 

.76 

.19 

.71 

•33 

3-04 

3-85 

4-75 

5-75 

6.84 

8.03 

9-3i 

10 

•05 

.20 

•45 

.80 

•25 

.80 

•45 

3.20 

4-05 

5-oc 

6.05 

7.20 

8-45 

9.80 

ii 

.06 

.22 

•49 

.88 

.40 

.98 

.69 

3-52 

4-45 

5-50 

6.65 

7.92 

9.29 

10.78 

12 

.06 

.24 

•  54 

.96 

•50 

.16 

•94 

3-84 

4.86 

6.00 

7.26 

8.64 

10.14 

11.76 

13 

.07 

.26 

•59 

.04 

-63 

•34 

3.i8 

4.16 

5-27 

6.50 

7-87 

9.36 

10.99 

12.74 

14 

.07 

.28 

•63 

.12 

•  75 

•  52 

3-43 

4.48 

5.67 

7.00 

8.47 

O.IO 

11.83 

I3-72 

15 

.08 

•30 

•67 

.20 

.87 

.70 

3.67 

4.80 

6.07 

7-50 

9.08 

0.80 

12.67 

14.70 

16 

.08 

•32 

.72 

.28 

.00 

.88 

3-92 

5-12 

6.48 

8.00 

9.68 

1.52 

13.52 

15-68 

17 

.OQ 

•  34 

-76 

.36 

.12 

3-o6 

4.16 

5-44 

6.88 

8.50 

10.28 

2.24 

14.36 

16.66 

18 

.Op 

.36 

.81 

.44 

•25 

3-24 

4.41 

5.76 

7.29 

9.00 

10.89 

2.96 

15-21 

17.64 

19 

.10 

.38 

-85 

•52 

•  37 

3-42 

4-65 

6.08 

7.69 

9-50 

11.49 

3-68 

16.05 

18.62 

20 

.10 

.40 

.90 

1.  60 

2.50 

3-6o 

4-90 

6.40 

8.10 

10.00 

12.10 

14.40 

16.90 

19.60 

TABLE  VIII. 
AZIMUTH  OF    POLARIS  WHEN    AT    ELONGATION. 


3 

1903. 

1904. 

1905. 

1906. 

1907. 

1908. 

1909. 

1910. 

1911. 

rt 

8° 

12° 

I"   14' 

•3 

°I4; 

:-; 

Ill', 

:-; 

•S> 

0  12' 

0  11' 
0  1  2' 

4° 
8° 

i6w 

0  is7 

°    IS' 

°  is' 

°  14' 

0  14' 

0  14' 

°I3' 

0  13' 

1  6° 

20° 

I°Ti7' 

0  I7/ 

°    I7' 

0  16' 

0  1  6' 

0  16' 

0  15' 

°  IS7 

°  I4/ 

20° 

24° 

«2?' 

°20' 

°2^ 

^ 

°I9' 

°i8; 

°i8; 

24° 

30° 
32° 

1°   26' 

°25' 

°25' 

°24' 

°24' 

°24' 

°23/ 

0  23' 

°  23' 

30° 

32° 

34° 

1°   28' 

°27/ 

°27/ 

0   26' 

0   26' 

0  26' 

025' 

°25' 

°  25' 

36" 

i°3o' 

0  29' 

°29' 

°29' 

°28' 

0  28' 

°27' 

°27' 

°27' 

36° 

38° 

I°32' 

1°  32' 

o31' 

o  ^j' 

°3o' 

3t°3'^ 

°2Q' 

I°29' 

38° 

40° 

i°  35' 

1*34' 

°34' 

I034' 

°33' 

i°  33' 

I°32/ 

°  32' 

I°32 

40° 

i03§' 

1*37' 

o37' 

I037' 

i°36' 

i°35' 

°  35' 

i°34 

42° 

44° 

1°  41' 

i°  41' 

°40' 

l°4o' 

°39' 

i°39/ 

i03«' 

°38' 

i°37 

44° 

46° 
48° 

i°45/ 
i°49/ 

i°44' 
i°48' 

i°44r 

I°48' 

I043' 
i°47' 

°43' 
°47' 

i°42' 
IJ.46J 

I°42' 

i°46/ 

i°4S' 

i°4i 
T°45 

46° 
48° 

50° 

i°53' 

i°53' 

I°52' 

I°52/ 

i°5o' 

I0.  50' 

i°49/ 

50° 

44 


TABLE  IX. 

TANGENT    DISTANCES,   LONG  CHORDS,  AND    EX- 
TERNAL  SECANTS  OF  A  ONE-DEGREE  CURVE. 

Also  amounts  to  be  added  to  same  when  No.  5,  No.  9, 
and  No.  14  Spirals  are  to  be  used. 

To  find  corresponding  functions  of  any  other  curve  divide 
tabular  number  by  degree  of  curvature. 

Formulas  used  in  construction  of  table: 

Long  chord   =  2X5729.58X8^0, 
Tangent         =5729.58  Xtan  a; 
External  sec  =  5729.58  -i-cos  0  —  5729.58; 
where  a  =  J  whole  angle  of  curve. 

With  Metric  System. 

Using  20-meter  chords,  divide  tabular  quantities  by  five 
times  the  proposed  metric  degree  of  curve  for  corresponding 
quantities  in  metric  measure. 

This  table  is  exact  for  arc  measurements  and  is  as  accurate 
as  any  table  like  it  for  the  chord-measurement  method  for 
laying  out  curves. 

45 


IX.— FUNCTIONS    OF   A   ONE-DEGREE   CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

1° 

50.0 

100  .0 

O  .  22 

1° 

2' 

5I-67 

103.3 

0.24 

2' 

4'' 

53-33 

106  .  7 

0.25 

4' 

6' 

55-0° 

IIO.O 

0,27 

6' 

8' 

56.67 

II3-3 

0.28 

8' 

10' 

58.33 

116  .7 

0.30 

10' 

12' 

60  .00 

I2O  .O 

0.31 

12' 

14' 

61.67 

123.3 

0-33 

14' 

16' 

63-33 

126.7 

0-35 

16' 

18' 

65.00 

130.0 

0-37 

18' 

20' 

66  .67  *o  o  ^ 

133-3 

0.39  v;°; 

20'' 

22' 

68.33  *"* 

136.7 

22r 

24' 

70-00^::  5 

140.0 

°-43    %~  " 

24' 

26' 

71-67  a 

143.3 

0-45     a 

26' 

28' 

73-34  J:  - 

146.7 

0-47     |:  r 

28' 

30' 

75.00*8 

150.0 

0.49    *§ 

30' 

32 

76.67.3.  : 

153-3 

0.51     |:  : 

32' 

34' 

78.34^ 

156.7 

0-54     +> 

34' 

36' 

8o.oo.S:  s 

160  .0 

o  .  56    -*s  : 

36' 

38' 

81.67  1^  _ 

163-3 

0.58   *§ 

38' 

40' 

83.34!:: 

166.7 

0.60   ^"  " 

40' 

42' 

170.0 

0.63     £ 

42' 

44' 

86!67  1 

173-3 

0.65      §. 

44' 

46' 

88.  34  1s  s 

176.7 

0.68     |"' 

46' 

48' 

90.01  * 

180.0 

0.71     - 

48' 

50' 

91.68^=  = 

183.3 

o.73    ^  " 

50' 

52 

93-34*8.  [ 

186.7 

0.76    -g.  „ 

52 

54' 

95.01  3~  " 

190.0 

o.79    3"" 

54' 

56', 

96.  68  .2:  = 

193-3 

0.81    .S2:  = 

56' 

58' 

98.34  £*? 

196.7 

0.84    "°"2 

O--    - 

58' 

2° 

100  .01  *" 

200  .0 

0.87  *- 

2° 

2' 

101  .67  '§- 

203.3 

0.90    'g 

2' 

4' 

103.34^  = 

206.6 

0.93  ^  = 

4' 

6' 

105    .01        JH 

210.0 

0.96     c 

6' 

8' 

I06.68|:  : 

213.3 

0.99     |:  s 

8' 

10' 

108.35^ 

216.6 

I  .02      " 

10' 

12' 

no.  02 

220  .0 

I  .06 

12' 

14' 

1  1  1.  68 

223.3 

I  .09 

14' 

16' 

113-35 

226.6 

I  .12 

16' 

18' 

115.01 

230.0 

1  •I5 

18' 

20' 

116.68 

233.3 

1  .19 

20' 

22' 

118.35 

236.6 

1.23 

22' 

24' 

I2O  .02 

240  .0 

1.26 

24' 

.26' 

121.68 

243-3 

1.30 

26' 

28' 

123.35 

246.6 

i-33 

28' 

46 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

2°  30' 

125  .02 

250.0 

1.36 

2°  30' 

V' 

126  .69 

253-3 

•39 

32' 

34' 

128.35 

256.6 

-43 

34' 

36' 

130.02 

260  .0 

•47 

36' 

38' 

131.69 

263.3 

•51 

38' 

40' 

I33-36 

266.6 

-55 

40' 

42' 

I35-03 

270.0 

-59 

42' 

44' 

136.70 

273.3 

1.63 

44' 

46' 

138.36 

276.6 

1.67 

46' 

48' 

140.03 

280.0 

1.71 

48' 

5°: 

141.70 

283.3 

i«75 

S°' 

52 

143.36 

286.6 

1.79 

5*' 

54' 

I45-03 

290.0 

1.83 

54' 

56,' 

146  .  70 

293-3 

1.88 

56' 

58' 

148.37 

296.6 

i  .92 

58' 

3° 

150  .04 

300.0 

i  .96 

3° 

2' 

JS1-?1 

303.3 

2  .OI 

2' 

4' 

153.38 

306.6 

2  .05 

4' 

6' 

155-04 

309.9 

2  .09 

6' 

8' 

156.71 

313.3 

2.14 

8' 

10' 

158.38 

316.6 

2  .  19 

10' 

12' 

160.05 

3I9-9 

2  .24 

12' 

14' 

161  .72 

323.3 

2  .  29 

14' 

16' 

163.38 

326.6 

2.34 

16' 

18' 

165.05 

329.9 

2.38 

1  8' 

20' 

166.72 

333.3 

2.43 

20' 

22' 

168.38 

336.6 

2.48 

22' 

24' 

170.05 

339-9 

2.52 

24' 

26' 

171.72 

343-3 

2-57 

26' 

28' 

173-39 

346.6 

2  .62 

28' 

30' 

175.06 

349.9 

2  .67 

30' 

32' 

176.73 

353-3 

2  .  72 

32' 

3< 

178.40 

356.6 

2.77 

34' 

36' 

180  .07 

359-9 

2.82 

36' 

38' 

181.74 

363.3 

2.87 

38' 

40' 

183.40 

366.6 

2.93 

40' 

42' 

185.07 

369.9 

2.98 

42' 

44' 

186.74 

373-3 

3«°4 

44' 

46' 

188.40 

376.6 

3.10 

46' 

48' 

190.07 

379-9 

3.15 

48' 

50' 

191.74 

383-3 

3.21 

S°' 

5^ 

193.40 

386.6 

3.26 

52' 

54' 

!95-o7 

389-9 

3.32 

54' 

56' 

196.74 

393-3 

3.38 

56' 

58' 

198.41 

396.6 

3.44 

58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle  •. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

4° 

200.08 

399-9 

3-49 

4° 

2' 

201.75 

403-3 

3.55 

2' 

4' 

203.42 

406  .6 

3.6l 

4' 

6' 

205  .09 

409.9 

3.67 

6' 

8' 

206.76 

4i3«3 

3-73 

8' 

10' 

208.43 

416  .6 

3-79 

10' 

12' 

2IO  .  IO 

419.9 

3-85 

12' 

14' 

211.77 

423-3 

3-92 

14' 

16' 

213-43 

426.6 

3.98 

16' 

18' 

215.10 

429.9 

4.04 

18' 

20' 

216.77 

433-3 

4.  10 

20' 

22' 

218  .44 

436.6 

4.  16 

22'. 

24' 

22O  .  II 

439-9 

4.  22 

24' 

26' 

221  .78 

443-2 

4.28 

26' 

28' 

223.45 

446  .6 

4.35 

28' 

30' 

225  .  12 

449-9 

4.42 

30' 

32' 

226.79 

453-2 

4.48 

32' 

34' 

228.46 

456.6 

4-55 

34' 

36' 

230.13 

459-9 

4  .62 

36' 

38' 

231  .80 

463.2 

4.69 

38' 

40' 

233-47 

466.6 

4.76 

40' 

42' 

235.14 

469.9 

4.82 

42' 

44' 

236.81 

473-2 

4.89 

44' 

46' 

238.48 

476.6 

4.96 

46' 

48' 

240.15 

479-9 

5-°3 

48' 

5°' 

241  .81 

483.2 

5.10 

50' 

52' 

243.48 

486.5 

5-17 

52/ 

54' 

245  -IS 

489.9 

5-24 

54' 

I*' 

246.82 

493-2 

5-3i 

56' 

#' 

248.49 

496.5 

5-38 

58' 

5° 

250.  16 

499-9 

5-46 

5° 

a' 

251-83 

503-2 

5-53 

2' 

4' 

253-50 

506.5 

5.60 

4' 

6' 

255-i7  ^ 

509-9 

5-68  ^ 

6' 

8' 

256.84  I- 

5I3-2 

5-75  6 

8' 

10' 

258.51  -g 

5*6-5 

5'832 

10' 

12' 

260.18  •§, 

5*9-9 

5-9o| 

12' 

14' 

261.85  <£ 

523-2 

5  -98  & 

14' 

16' 

263.52  £ 

526.5 

6.06  g 

16' 

18' 

265.19    o- 

529.8 

6.13^ 

18' 

20' 

266.86  « 

533-2 

6.21    ^ 

20' 

22' 

268.53  3 

536.5 

6.293 

22' 

24' 

270.20  ^ 

539-8 

6.37 

24' 

26' 

271.87 

543-2 

6.45 

26' 

28' 

273-54 

546.5 

6.53 

28' 

48 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

5°  30' 

275.21 

549-8 

6.61 

5°  30' 

32' 

276   88 

553-2 

6.69 

32' 

34' 

278    56 

S56-5 

6.77 

34' 

36; 

280.23 

559-8 

6-85  ± 

36' 

38' 

281  .  90 

563-2 

6-93   6 

38' 

40' 

283-57 

566.5 

7-0x5 

40' 

42' 

285.24 

569-8 

7-o9.§ 

42' 

44' 

286.91 

573-2 

7  •  J7  co! 

44' 

46' 

288.58 

576.5 

7-25   o 

46' 

48' 

290.25 

579-8 

7-34  £ 

48' 

50' 

291.92 

583-1 

7-43  | 

50' 

52' 

293-59 

586.5 

52' 

54 

295  .  26 

589.8 

7  .60 

54' 

56' 

296.93 

593-1 

7.68 

56' 

S8 

298  .60 

596.5 

7-77 

58' 

6° 

300  .  28 

599-8 

7-85 

6° 

2' 

301  -95 

603.1 

7-94 

2' 

4' 

303-63 

606.5 

8.03 

4' 

6' 

305-30     'o 

609.8 

8.12 

6' 

8' 

306.97  1 

613.  i 

8.21 

8'- 

10' 

308.6413 

616  .4 

8.30 

10' 

J27 

310  31  -a 

619.8 

8-39 

12' 

14' 

311.98   C£ 

623.1 

8.48 

14' 

16' 

3*3-66  .0 

626  .4 

8-57 

16' 

18' 

3J5-33   2 

629.7 

8.66 

r8' 

20' 

317.00  E 

633-1 

8-75 

20.' 

22' 

318.67  5 

636.4 

8.84 

22' 

24' 

320.34  < 

639.7 

8-94    . 

24' 

26' 

322  .01 

643.1 

9-03  £ 

26' 

28' 

323.68 

646  .4 

9-13  5zi 

28' 

3°' 

325.35 

649-7 

9-23  1 

30' 

32' 

327.02 

653-0 

9-32  $ 

32 

34 

328.70 

656.4 

9.42     £ 

34' 

36' 

330.37 

659-7 

9-52  S 

36' 

38' 

332-04 

663.0 

9  .62    N 

38' 

40' 

333.71 

666.3 

9.7i  | 

40' 

42' 

335-39 

669.7 

'       9.81  * 

42' 

44' 

337-o6 

673.0 

9.91 

44' 

46' 

338.73 

676.3 

IO  .OO 

46' 

48' 

340.40 

679.6 

IO  .  IO 

48' 

50'    - 

342.08 

$83-0 

IO  .  2O 

50/ 

52' 

343-75 

686.3 

10.30 

52^ 

54'  . 

345-42 

689.6 

10.40 

54 

56' 

347-io 

692  .9 

10.50 

56 

58' 

348.77 

696.3 

10  .60 

58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

7° 

350-44 

699.6 

10  .  71 

7° 

2' 

352.ii 

702.9 

io.il 

2' 

4' 

353.78 

706  .  2 

10  .91 

4' 

6' 

355-45 

7°9-5 

II  .01 

6' 

8' 

357-12 

712.9 

II  .  II 

8' 

10' 

358.8o 

7l6  .  2 

11.22 

10' 

12' 

360.47 

7T9-5 

11.32 

12' 

14' 

362  .14 

722.8 

n-43 

14' 

16' 

363-82 

726  .  2 

IJ-53 

16' 

18' 

365-49  ^ 

729-5 

ii  .  64 

18' 

20' 

367-I7   5° 

732-8 

n-75 

20' 

22' 

368.84^ 

736.1 

ii  .85 

22" 

24' 

370.51  .b 

739-5 

ii  .  96 

24' 

26' 

372.19  £ 

742.8 

12  .07 

26' 

28' 

373-86  £ 

746.1 

12  .18 

28' 

3°; 

375-54  ;• 

749-4 

12  .29 

30' 

32' 

377-21   ^ 

752.8 

12  .40 

32' 

34' 

378-88  ^ 

756.1 

12  .  51 

34' 

36' 

380.  56  ^ 

759-4 

I2-63   ^ 

36' 

38' 

382.23 

762.8 

12.74    6 

38' 

40' 

383-91 

766.1 

12.85^ 

40' 

42' 

385.58 

769-4 

I2.96| 

42' 

44' 

387-25 

772.7 

13.08  £ 

44' 

46' 

388.93 

776.1 

13-19    o 

46' 

48' 

390.61 

779-4 

13-30   o 

48' 

50' 

392.28 

782.7 

13.41    -d 

50' 

52' 

393-95 

786.0 

13.53   3 

52' 

54' 

395-62 

789.4 

13.64 

54' 

56' 

397-30 

792.7 

13.76 

56' 

58' 

398.97 

796.0 

13.88 

58' 

8° 

400  .65 

799-4 

13.99 

8° 

2' 

402.33 

802.7 

14.  10 

2' 

4' 

404.00 

806.0 

14.22 

4' 

6' 

405.67   -- 

809.3 

14.34 

6' 

8' 

407-35  g 

812.6 

14.46 

8' 

10' 

409.03  -g 

816.0 

14.58 

10' 

12' 

410.70  -a 

819-3 

14.70 

12' 

14' 

412.38^ 

822.6 

14.82 

14' 

16' 

414  .06  £ 

825.9 

14.94 

i6' 

1  8' 

415-74    d 

829.3 

15  .06 

18' 

20' 

417.41  5 

832.6 

15.18 

20' 

22' 

419.08  ? 

835.9 

15-30 

22' 

24' 

420.76  < 

839.2 

15-43 

24' 

26' 

422  .44 

842.6 

J5-55 

26' 

2.8' 

424  .  IT 

845-9 

T5  -67 

28' 

50 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

8°  30' 

425.79 

849.2 

15.80 

8°  30' 

32' 

427.47 

852.5 

15-93 

32' 

34' 

429.I5 

855-8 

16  .06 

34' 

36' 

430.83 

859.2 

16.  19 

36' 

38' 

432.50 

862.5 

16  .31 

38' 

40' 

434.17 

865.8 

16.43 

40' 

42' 

435.85 

869.1 

16.56 

42' 

44' 

437-53 

872.5 

16.69 

44' 

46' 

439.20 

875.8 

16.81 

46' 

48' 

440.88 

879.1 

16.94 

48' 

50' 

442.55 

882.4 

17.07 

50' 

5*' 

454-23 

885.7 

17  .20 

52' 

54' 

455-90 

889.  I 

17-33 

54' 

56' 

437-5S 

892.4 

17.46 

56' 

58' 

459.26 

895-7 

J7-59 

58' 

*>° 

450.93 

899  .0 

17.72 

9° 

2' 

452  .60 

902.3 

17-83 

2' 

4' 

454.28      u^a 

905-6 

17.97  ">» 

4' 

6' 

455^6    o*. 

909  .0 

18.10  d 

6' 

8' 

457-64* 

912.3 

18.26  *: 

8' 

io' 

459-32  •£: 

9I5-7 

i8.38.1: 

10' 

12' 

4f  i  .00  w 

919.0 

18.52  & 

12' 

14' 

4^2.68  .0: 

922.4 

18.65  J3: 

14' 

16' 

4^4.36   «•? 

925-7 

18.79  °  « 

16' 

18'. 

4^6.04  £& 

929.0 

18.93  ^r 

18' 

20' 

467-71  g. 

932.3 

19  .06    S; 

20' 

22' 

469.39  < 

935-6 

19.19    < 

22' 

24' 

471-07 

939-o 

*9'33 

24' 

26' 

472.75 

942.3 

19.49 

26' 

28' 

474.42 

945-6 

19.62 

28' 

3°' 

476.10 

948.9 

J9-75 

30' 

3^' 

477-78 

952.2 

19.89 

32' 

34' 

479.46 

955-5 

20.03 

34' 

36' 

481  .13 

958.9 

20.  17 

36' 

38' 

482.81 

962  .  2 

20.31 

38' 

40' 

484  .  49 

965.5 

20.45 

40' 

42' 

486.17 

968.9 

20.59 

42' 

44' 

487.84 

972.2 

20.74 

44' 

46' 

489.52 

975-5 

20.88 

46' 

48' 

491  .  20 

978.8 

21  .02 

48' 

5°' 

492.88 

982.1 

21  .  l6 

5°' 

52' 

494.56 

985-4 

21.30 

S^' 

54' 

496.24 

988.7 

21.45 

54' 

56' 

497.92 

992.0 

21.59 

56' 

58' 

499  .60 

995-3 

21    74 

58' 

51 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

10° 

501  .28 

998.7 

21  .89 

1O° 

2' 

502.96 

IOO2  .O 

22  .04 

2f 

4' 

504.64 

1005  .4 

22.18 

4' 

6' 

506.32 

I008.-7 

22.33 

6' 

8' 

508.0 

IOI2  .O 

22.48 

87 

10' 

509.68 

IOI5.3 

22  .63 

10' 

12' 

5H-36 

1018.  7 

22.78 

12' 

14' 

5*3-04 

IO22  .O 

22.93 

14' 

1  6' 

514-72 

I025-3 

23.08 

16' 

1  8' 

516  .40 

1028.6 

23.23 

18' 

to' 

518.08 

1031  .9 

23-38 

20' 

22' 

5I9-76 

1035.3 

23-53 

22' 

24' 

52i  -44 

1038.6 

23.69 

24' 

26' 

523-12 

1041  .  9 

23.84 

26' 

28' 

524.80 

1045.2 

23.99 

28' 

30' 

526.48 

1048.5 

24.14 

30' 

32/, 

528.16 

1051  .8 

24.30 

32; 

34 

529.84     xAo 

1055-2 

24-45 

34 

36 

53I-52    o- 

1058.5 

24.60    ^ 

36' 

38' 

533-20  « 

1061.8 

24-75    6. 

38' 

40' 

534-  88  4= 

1065  .  i 

24.91  * 

40' 

4V 

536.  #6  cc 

1068  .4 

25.06  |- 

42' 

44' 

538.4.4  £: 

1071.7 

25.22  c& 

44' 

46' 

540.CJ2     «t« 

1075  .0 

25-37    c: 

46' 

48' 

541.60     ££ 

1078.4 

25.54  *p 

48' 

50' 

543-29   ?- 

1081  .  7 

25.70  •£*" 

50' 

52 

544-97  < 

1085  .0 

25-86^ 

52' 

54' 

546.65 

1088.4 

26.02 

54 

56' 

548.33 

1091.7 

26.18 

56' 

58' 

550.02 

1095.0 

26.34 

58' 

11° 

55I-70 

1098.3 

26  .  ^o 

11° 

2' 

553.38 

noi  .6 

26.66 

2' 

4' 

555-o6 

1105.0 

26.83 

4r 

6' 

556.74 

1108.3 

26  .99 

6' 

8' 

558.43 

mi  .6 

27-I5 

8' 

to' 

560.11 

1114.9 

27-  31 

10' 

12' 

561.80 

1118.3 

27.47 

12' 

14' 

563-48 

II2I  .6 

27.64 

14' 

16' 

565-16 

1124.9 

27  .  80 

i6f 

18' 

566.84 

1128  .  2 

27.97 

18' 

20' 

568.53 

H3I.5 

28.14 

20r 

22' 

570.22 

II34.8 

28.30 

227 

24' 

571-9° 

;  1.138.1 

28.47 

24' 

26' 

573.58 

1141.4 

28.64 

26' 

28' 

575-26 

1144.7 

28.80 

28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

11°  3°' 

576.95 

1148  .0 

28.97 

11°  30' 

32' 

578  .  63 

1151.4 

29.14 

32' 

34' 

580.32 

ri54-7 

29.31 

34' 

36' 

582  .00    ^os 

1158.0 

29.48 

36' 

38' 

583.69  |: 

1161.3 

29.65 

38' 

40' 

585-37    g 

1164.6 

29.82 

40' 

42' 

587.05  '£: 

1168.0 

30.00 

42' 

44' 

588.73^ 

1171  .3 

30.17 

44' 

46' 

590.42  £- 

1174.6 

30.34 

46' 

48' 

592.io   2| 

1177.9 

30.51 

48' 

50' 

593-79   «« 

1181.2 

30.68 

50' 

52' 

595-47  S: 

1184.6 

30.85 

52/ 

54' 

597.16  < 

1187.9 

31.02 

5< 

56' 

598.84 

II9I  .2 

31  .  2O 

56 

58' 

600.53 

II94-5 

31-37 

58' 

12° 

602  .21 

II97.8 

3I-56 

12° 

2' 

603.89 

I2OI  .  2 

31-73 

2' 

4' 

605.58 

1204  .  5 

3L9I 

4' 

6' 

607  .  27 

1207  .8 

32.09    u^a 

6' 

8' 

608.96 

I2II  .  I 

32.27    6. 

8' 

.10' 

610  .64 

I2I4.4 

32.45  -g 

10' 

12' 

612.32 

I2I7.7 

12' 

14' 

614.01 

1221  .O 

32  .81  & 

14' 

1  6' 

615.70 

1224.3 

33-00  JD: 

16' 

1  8' 

617.38  .Ac* 

1227.6 

33.18  ^^ 

1  8' 

20' 

619.07   6- 

1230.9 

33-35  |," 

20' 

22' 

620.76  Ij 

1234.3 

33-53  < 

22' 

24' 

622  .45  .§-. 

1237.6 

33-71 

24' 

26' 

624.13     £ 

1240.9 

33.89 

26' 

28' 

625.82     |- 

1244.2 

34.07 

28' 

3°' 

627.50   *?  *r 

1247-5 

34.26 

30; 

32' 

629.19  aft 

1250.9 

34.44 

32 

34' 

630.87  ^ 

1254.2 

34.62 

34 

36' 

632-56  < 

1257.5 

34.80 

3£ 

38' 

634.24 

1260.8 

34-99 

38' 

40' 

635-93 

1264.  I 

35.18 

40' 

42' 

637.62 

1267.4 

35-36 

42' 

44' 

639-3° 

1270.7 

35-55 

44' 

46' 

640.09 

1274.0 

35-73 

46' 

48' 

642.68 

I277-3 

35-92 

48' 

5°' 

644.37 

1280.6 

36.12 

5°' 

52' 

646  .06 

1284.0 

36.31 

52' 

54' 

647-75 

1287.3 

36.5° 

54 

56' 

649.44 

1290  .6 

36.69 

56 

58' 

651-13 

1293.9 

36.88 

58' 

53 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance: 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

13° 

652.81 

1297.2 

37.07 

13° 

2' 

654-50 

1300.5 

37.26 

2' 

4' 

656.  19 

1303-8 

37-45 

4' 

6' 

657.88 

1307.1 

37-64 

67 

8' 

659-57 

1310.4 

37.83 

8' 

10' 

661  .25 

!3i3.7 

38.03 

10' 

12' 

662  .94 

1317.0 

38.22 

12' 

14' 

664-63 

1320.4 

38.41 

14' 

16' 

666  .32 

1323.7 

38.60 

16' 

18' 

668.01    .  . 

»O  O\ 

1327.0 

38.80 

18' 

20' 

669  .  70  6, 

!330-3 

39-00  ^ 

207' 

22' 

671-395" 

J333.6 

33.19^ 

22' 

24' 

673.08!, 

1336.9 

39-391 

24' 

26' 

674.77c&' 

1340.2 

39-59;^ 

9# 

28' 

676.46^. 

1343-5 

39-79^ 

28' 

3o' 

678.15^ 

1346.8 

39-99^ 

30' 

32' 

679.84  ££ 

1350-2 

40  .  19  N  *- 

32' 

34' 

681.53^ 

1353.5 

40-3955 

34' 

36' 

683.22^- 

1356.8 

40.59^ 

36' 

38' 

684.91 

1360.1 

40.79 

38' 

40' 

686.60 

1363-4 

40.99 

40' 

42' 

688.29 

1366.7 

41.19 

42' 

44' 

689.98  . 

1370.0 

.41-39 

44' 

46' 

691  .67 

J373-3 

4L59 

46' 

48' 

693.36 

1376.6 

41  .80 

48' 

50' 

695-05 

J379-9 

42  .00 

50' 

52' 

696.74 

1383-3 

42  .  20 

52' 

54' 

698.43 

1386.6 

42.40 

54' 

56' 

2fOO  .  12 

1389.9 

42  .61 

56' 

58' 

^OI  .8l 

1393-2 

42.82 

58' 

14° 

703.5I 

1396.5 

43.03 

14° 

2' 

705.20 

1399.8 

43.24 

2' 

4' 

706  .89  >"  o*4 

1403.1 

43-44  ,^4 

4' 

6' 

708.58  6(  ; 

1406  .4 

43.65    .     M 

6' 

8' 

710.27  £~  " 

1409.7 

43-86^:: 

8X 

to' 

7"-  97  |:  = 

1413.0 

44.07  ^ 

10' 

12' 

7i3.66c& 

1416.4 

44.27  ;a:  : 

I2X 

14' 

715-36^:  : 

1419.7 

44-48^ 

14' 

1  6' 

717.05    ^oo  H 

1423.0 

44.7o^:  = 

1  6' 

18' 

718.74  aas 

1426.3 

44.91  trZo 

18' 

20' 

720.43^ 

1429.6 

45.12  g 

20' 

22'  - 

722.12  ^!  • 

1432.9 

45-33<"  : 

22' 

24' 

723.81 

1436.2 

45-54 

24' 

26' 

725.50 

1439.5 

45-75 

26' 

2g' 

727.20 

1442  .8 

45-97 

28' 

54 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance 

Long 
Chord. 

External 

Secant. 

Whole 
Angle. 

14°  30' 

728  .90 

1446.  I 

46.18 

14°  30' 

32' 

730-59 

1449-5 

46  .40 

3^' 

34' 

732  .28 

1452  .8 

46  .61 

34' 

36' 

733-98 

1456.1 

46.83 

36' 

33' 

735-67 

1459-4 

47.04 

38' 

40' 

737-37 

1462.7 

47-25 

40' 

42' 

739.06 

1466  .0 

47-46 

4=' 

44' 

740.76 

1469.3 

47-68 

44' 

46' 

742.45 

1472.6 

47.90 

46' 

48' 

744-15 

1475-9 

48.12 

48' 

5°' 

745  -84 

1479.2 

48.34 

5°' 

52' 

747-54 

1482.5 

48.56 

S^' 

54' 

749-23 

1485.9 

48.78 

54' 

56' 

750-93 

1489.2 

49  .00 

56' 

58' 

752  .62 

1492.5 

49.22 

58' 

15° 

754.32 

1495.8 

49.44 

15° 

2' 

756.02 

1499.1 

49.68 

2' 

4' 

757-71  «Aont 

1502.4 

49-90  ^^^ 

4' 

6' 

759-40  6^  ^ 

I5°5-7 

5°"I3 

6' 

8' 

761  .  10  £•  • 

1509  .0 

5°-34£:  : 

8' 

10' 

762.80  J_  , 

i5I2-3 

SO-SS'g 

10' 

12' 

764.49'd"  " 

I5I5-6 

50.77  •&=  = 

12' 

14' 

766.18  fc.  5 

1518.9 

51.00* 

14' 

16' 

767.  88  -„„ 

1522  .  2 

51.23^-  : 

16' 

1  8' 

769-58  OOH 

I525-5 

1:1  AC;  ^^^" 

S*    'T-J      W    0*C 

18' 

20' 

_  «  •»!-  i> 
771.28.3 

1528.8 

51.67^ 

20' 

22' 

772.983- 

1532.1 

5i«9o<-  - 

22' 

24' 

774-68 

1535.4 

52.14 

•  24' 

26' 

776.37 

1538.7 

52.37 

26' 

28' 

778.07 

1542.0 

52  .60 

28' 

3°' 

779-77 

1545.3 

52.82 

3°' 

32' 

781.47 

1548.6 

53-05 

32 

34' 

783-17 

i55i-9 

53-30 

34' 

36' 

784.86 

1555-2 

53-53 

36' 

38' 

786.56 

1558-5 

53.75 

38' 

40' 

788.26 

1561.8 

53-97 

40' 

42' 

789.96 

1565-1 

54-20 

42' 

44' 

791  .66 

1568.4 

54-43 

44' 

46' 

793-35 

i57J-7 

54.67 

46' 

48' 

795-05 

i575-o 

54-90 

48' 

SO'' 

796.75 

1578.3 

55-J3 

5°' 

S^' 

798.45 

1581.6 

55-36 

S^' 

54' 

800  .  15 

1584.9 

55-6o 

54 

56' 

801  .85 

1588.2 

55-84 

56 

58' 

803.55 

i59i.5 

56-08 

58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

16  ° 

805.25 

1594.8 

56.31 

16° 

2' 

806.95 

1598.1 

56.55 

2' 

4' 

808.65 

1601  .4 

56.80 

4' 

6' 

810.35 

1604.  7 

57.03 

6' 

8' 

812  .  05 

1608  .0 

57.27 

8' 

10' 

813.75 

1611.3 

57-5° 

10' 

12' 

815-45 

1614  .6 

57-74 

12' 

14' 

817.15 

1617.9 

57-98 

14' 

16' 

818.85 

l62I  .2 

58.22 

16' 

18' 

820  55 

1624.5 

58.46 

18' 

20' 

822  .25         4 

1627.8 

58-70 

20' 

22' 

823.95          6 

1631  .1 

58.95 

22' 

24' 

825.65         £ 

1634.4 

59.20 

24' 

26' 

827.35         1 

1637.7 

59-43 

26' 

28' 

820.06        'a 
a: 

1641  .O 

59.67 

28' 

3o' 

830.76         £ 

1644.3 

59-91 

30' 

32' 

832.46         ^ 

1647  -6 

60  .  16 

32' 

34' 

834-16    .  .5 

1650.9 

60.40  ^4 

34' 

•      36' 

835.86--- 

1654.2 

60.65    • 

36' 

38' 

S37.56£:| 

J657-5 

60  .90  ^:  - 

38' 

40' 

839.27?. 

1660.8 

61  .  14  ^ 

40' 

42' 

840.  97  |T 

1664.  i 

6l-39|<:  " 

42' 

44' 

842  .67  t^ 

1667  .4 

61  .64  v. 

44' 

46' 

844.  3  7^0  . 

1670.7 

61.89^^ 

46' 

48' 

846  .  07   6  6  -t 

1674  .0 

62   .   14     oi    t^vc' 

48' 

50' 

847-78^ 

1677-3 

02  .38  ^ 

50' 

52' 

849.483-  5; 

1680.6 

62  .63  <T 

52' 

54' 

851.  18^  | 

1683.9 

62.88 

54' 

56' 

852.88     £ 

1687.2 

63-13 

56' 

58' 

854.59     j 

1690.5 

63.38 

58' 

17° 

856.30 

1693.8 

63-63 

17° 

2' 

858.00           o 

1697.1 

63.88 

2' 

4' 

859.70      £ 

1700  .4 

64.13 

4' 

6' 

861.41      5 

1703-7 

64-38 

6' 

8' 

863.11 

1707.0 

64.64 

8' 

10' 

864.82 

1710.3 

64  .  90 

10' 

12' 

866.52 

1713.6 

65.16 

12' 

14' 

868.23 

1716.9 

6s  .42 

14' 

16' 

869.93 

i  7^o  .  2 

65.68 

16' 

18' 

871.64 

1723-  4 

65-93 

18' 

20' 

873-35 

1726.7 

66.18 

20' 

22' 

875-05 

1730.0 

66.43 

22' 

24' 

876.76 

1733-3 

66.68 

24' 

26' 

878.46 

1736.6 

66.95 

26' 

28' 

880.  17 

1739-9 

67.23 

28' 

56 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

17°  30' 

881.88 

*743-2 

67.47 

17°  30' 

32' 

883.58 

1746.5 

32' 

34' 

885.29        .    . 

1749.8 

68.00 

34' 

36' 

886.99       a? 

*753-* 

68.25 

36' 

38' 

888.70       c. 

1756.4 

68.50 

38' 

40' 

890.41      -g 

*759-7 

68.77 

40' 

42' 

892  .11      •£- 

1763.0 

69.03 

42' 

44' 

893.82      * 

1766.2 

69-30 

44' 

46' 

895.53      £: 

I769-5 

69-56 

46' 

48r 

897.24       ?<•>. 

1772.8 

69.82 

48' 

50' 

898:95       3R 

1776.1 

70.09 

5°' 

52' 

900.66      ?- 

1779.4 

70.36 

52' 

54' 

902-37      "*' 

1782.7 

70.63 

54' 

56' 

904.08 

1786.0 

70.90 

56 

58' 

905-79 

1789.3 

71.17 

58' 

18° 

907.49 

1792.6 

7*  -42 

18° 

2'* 

909  .  2O 

1795-9 

71.68 

2' 

4' 

910.91           4 

1799.2 

7*-95 

4' 

6' 

912  .62 

1802  .5 

72.22  iA  £  4 

6' 

8' 

9*4-33        & 

1805.8 

72.49  0-_  f 

8' 

TO' 

916.03        ** 

1809.  I 

72  .  76  ^ 

10' 

12' 

9*7-74        $ 

1812  .4 

73  .  03  ^.^ 

12' 

14' 

9*9-45         fc 

I8I5.7 

73.30^  " 

14' 

16' 

921  .  16 

1819.0 

73.58  fe:  . 

16' 

18' 

922.87 

1822.2 

73.86^ 

18' 

20' 

924.5810- 

1825-5 

74.12  ^^ 

20' 

22' 

926  .  29  I2T  J2 

1828.8 

74.40?,  , 

22' 

24' 

928.00^ 

1832.1 

74.67^ 

24' 

26' 

929.71  -a: 

I835.4 

74.94 

26' 

28' 

931-42^ 

1838.7 

75.22 

28' 

30' 

933.13  ^~0 

1842  .0 

75-49 

30' 

32' 

934.84  c  ~4 

1845-3 

75-76 

32/ 

34' 

936.55   *W  6 

1848.6 

76.03 

34' 

36' 

938.26  -0:   ^ 

1851.9 

76.30 

36' 

38' 

939-  98  <      § 

1855.2 

76-58 

38' 

40' 

941-69         £ 

1858.5 

76.87 

40' 

42' 

943.40         J5 

1861.8 

77   *5 

42' 

44' 

945  •*  *        « 

1865.1 

77-43 

44' 

46' 

946.82           £ 

1868.4 

77-7o 

46' 

48' 

948.53          £ 

1871.7 

77.98 

48' 

50' 

950.25          3 

1875-0 

78.26 

50' 

52' 

95L96 

1878.3 

78.55 

52' 

54'. 

953  -67 

1881.6 

78.84 

54' 

56' 

955-39 

1884.9 

79-*3 

56' 

58' 

957-10 

1888.2 

79.40 

58' 

57 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

19° 

958.81 

1891  .4 

79.67 

19° 

2' 

960.52 

1894.7 

79-95 

2> 

4' 

962.23 

1898.0 

80  .  24 

4' 

6' 

963-95 

1901  .  2 

80.52 

6' 

8' 

965.67 

1904.5 

80.80 

8' 

10' 

967.38 

1907  .8 

81  .09 

10' 

12' 

969.  10 

1911  .  i 

8i.37 

12' 

14' 

970.82 

1914.4 

81.66 

14' 

16' 

972.53 

1917.7 

81.95 

16' 

i$f 

974.24 

1921  .0 

82.23 

18' 

20' 

10  Qv  TJ- 

975-95 

1924  .  2 

82.52 

20' 

22' 

977.66£:   : 

I927-5 

82  .80 

22' 

24' 

979-38- 

1930  .8 

83.09 

24' 

26' 

981.09.^:  : 

1934.0 

83-38 

26' 

28' 

982.81  & 

1937-3 

83-67 

28' 

30' 

984-  53  &  : 

1940.6 

83.97 

30' 

32' 

986.24  r~* 

1943-9 

84.26 

32' 

34 

987.96  £3° 

1947.2 

84-55 

34' 

36' 

989-675.  . 

!950-5 

84  •  84     *r>  O*  *t 

36' 

38' 

99*-39< 

1953-8 

85.13     6_W 

38' 

40' 

993  -11 

I957-1 

85.43S"~ 

40' 

42' 

994.83 

1960.4 

85-72  |:  , 

42' 

44' 

996.55 

1963  .6 

86.oi£ 

44' 

46' 

998  .  26 

1966  .9 

86.30  fc.  . 

46' 

48' 

999.98 

1970.2 

B6v«6--5^u 

48' 

50' 

1001  .  70 

1973-5 

86.90  "^ 

50' 

52' 

1003.42 

1976.8 

87-20^,  : 

52' 

54' 

1005.13 

1980.  i 

87.50^ 

54' 

56' 

1006.85 

1983-3 

87.80 

56' 

58' 

1008  .  56 

1986.6 

88.10 

58' 

2O° 

1010.3 

1989.9 

88.39 

2O° 

2' 

IOI2  .  I 

1993.2 

88.69 

2' 

4' 

IOI3.8       ^j 

1996.5 

88.99 

4' 

6' 

IOI5-5      6-    6 

1999.7 

89.29 

6' 

8' 

1017.3     £"  ^ 

2003  .0 

89.59 

8' 

10' 

1018.9  |=  4 

2006  .3 

89.89 

10' 

12' 

1020.6    <n    w 

2009  .6 

90.19 

12' 

14' 

1022  .3      J3:   J5 

2012  .9 

90.49 

14' 

16' 

1024.0     ^l  q 

2Ol6  .  I 

90.80 

16' 

18' 

iO2c;  .7     S,^  S 

0      '         cV^-R 

2OI9  .4 

91  .  10 

18' 

20' 

1027.5  5^ 

2O22  .  7 

91.40 

20' 

22' 

1029.2    < 

2026  .  o 

91.70 

22' 

24' 

1031  .0 

2029.3 

92  .00 

24' 

26' 

1032.7 

2032.5 

92.30 

26' 

28' 

1034.4 

2035.8 

92  .61 

28' 

58 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

20°  30' 

1036.1 

2039.1 

92  .92 

20°  30' 

32' 

1037.8 

2042  .4 

93-23 

32' 

34' 

1039-5 

2045-7 

93-53 

34' 

36' 

1041.3 

2049  .0 

93-84 

36' 

38' 

1043.0 

2O52  .  2 

94-15 

38' 

40' 

1044.7 

2055-5 

94.46 

40' 

42' 

1046.3 

2058.8 

94-77 

42' 

44' 

1048  .0 

2O62  .  I 

95-o8 

44' 

46' 

1049.7 

2065  .4 

95-40 

46' 

48' 

1051-5 

2068.6 

95-71 

48' 

So' 

1053-3 

2071.9 

96.02 

5^ 

$*' 

1055.0 

2075.2 

96.34 

52' 

54' 

1056.7 

2078.5 

96.66 

54' 

56' 

1058.4 

2081  .7 

96.97 

56' 

58' 

IO6O  .  2 

2085  .O 

97.28 

58' 

21° 

1061  .9  . 

2088.3 

97.58 

21° 

2' 

1063  .6 

2091  .6 

97.90 

2' 

4' 

1065.3    . 

2094.9 

98.22 

4' 

6' 

1067.0  1*°? 

2098.  i 

98.54  ^c-,4 

6' 

8' 

1068.8  d:  : 

2101  .4 

98.85  0-.  : 

8' 

10' 

I07o.5^ 

2104.7 

99.16^ 

10' 

12' 

1072.2.*:  : 

2IO8  .O 

99-47  -i=:  : 

12' 

1.4' 

1073.  9co 

2III  .  2 

99  -78  co 

14' 

16' 

1075.7.0=  = 

2II4.5 

100.09  J5:  : 

16' 

18' 

I077-4  *p;j 

2II7.8 

IOO  .  42  o  f^o 

18' 

20' 

o 

1079.  1  ££t- 

2121  .  I 

100.75    *^ 

20' 

22' 

1080.8^,  : 

2124.4 

IOI  .06  *d:  : 

22' 

24' 

1082.  5<r  " 

2127.7 

101.38* 

24' 

26' 

1084.3 

2131  .0 

101  .  70 

26' 

28' 

1086  .0 

2134.2 

IO2  .02 

28' 

30' 

1087  .8 

2*37-5 

102.35 

30; 

32' 

1089.5 

2140  .8 

102  .67 

32 

3< 

IO9I  .  2 

2144.1 

103  .00 

34 

36 

I  093rO 

2147.4 

103.32 

36 

38' 

1094.7 

2150  .6 

103.64 

38' 

40' 

1096  .4 

2153-9 

103.97 

40' 

42' 

1098.  I 

2157.2 

104.30 

42' 

44' 

1099.9 

2160  .4 

104.62 

44' 

46' 

noi  .6 

2163.7 

104.94 

46' 

48' 

1103.3 

2167  .0 

105.27 

48' 

5°; 

1105.1 

2I7O  .  2 

IO5  .60 

5°; 

52 

1106  .  9 

2173-5 

105.93 

52 

54' 

1108.6 

2176.8 

106  .  26 

54 

56' 

1110.3 

2l8o  .O 

106  .60 

56' 

58-' 

III2  .O 

2183.3 

106   92 

& 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

22° 

1113.7 

2186.6 

IO7  .2 

22° 

2' 

1115.4 

2189  .9 

107.5 

2' 

4' 

III7.2             ^        2193  .  2 

107.9 

4' 

6' 

IIl8.9             ^         2196.4 

108.2 

6' 

8' 

1120  .6        ^      2199  .  7 

108.6 

8' 

10' 

1122.3       *£ 

2202  .9 

108.9 

10' 

12' 

1124.0        '5. 

22O6  .  2 

109.3 

12' 

14' 

1125  .8         C      2209  .  5 

109  .6 

14' 

16' 

1127.5        £      2212.7 

no  .0 

16' 

18' 

1129.2         „      2216.0 

o    ! 

110.3 

18' 

20' 

2219.3 

no  .6 

20' 

22' 

II32.7           | 

2222  .6 

110.9 

22' 

24' 

II34-5 

2225.8 

111.3 

24' 

26' 

II36.2 

2229  .  I 

in  .6 

26' 

28' 

II38.0 

2232  .4 

112  .O 

28' 

30/ 

H39-7 

2235.6 

112.3 

3°' 

32' 

1141  .3 

2238.9 

112  .6 

32' 

34r 

1143.0  ^&* 

2242  .  I 

113.0 

34' 

36' 

1144.8  0-_  , 

2245.4 

113-3  ^dz 

36' 

38' 

1146.6  £~ 

2248  .  7 

II3-7   6.  : 

38' 

40' 

1148  .4  .§:  ; 

o<~ 

2252  .0 

114-0^"    " 

40' 

42' 

1150  .  i  co 

2255.3 

114  .  3  .£;    ^ 

42' 

44' 

1151  .80:: 

2258.5 

114.6  co 

44' 

46' 

1153    .6      U-,    Tj-10 

2261.8 

II5.°,0:    = 

46' 

48' 

ii55-3  ££S 

2265  .  i 

II5.3  o  tvq 

48' 

5o; 

1157.0  -d  * 

2268.3 

11$.  J    " 

50' 

II58.83J2  " 

2271  .  6 

1  16  .  o  »o-  ^ 

52' 

54' 

1160  .  5 

2274.9 

116.4^ 

54' 

56' 

1162.3 

2278.  2 

116.  7 

56' 

58' 

1164.0 

2281.5 

117.0 

58' 

23° 

1165.7 

2284.7 

117.4 

23° 

2' 

1167.4 

2288.0 

117.7 

2' 

4' 

ii$9  .  i         * 

2291  .  2 

118.1 

4' 

6' 

1190.9        c 

2294.5 

118.4 

6' 

8' 

11^2.6           ^ 

2297.7 

118.7 

8' 

10' 

H74.3        -g, 

2301  .0 

119.1 

10' 

12' 

1176  .0         co 

2304.3 

119.4 

12' 

14' 

1177.8         ,0 

2307.5 

119.7 

14' 

1  6' 

1179.6 

2310.8 

I2O  .  I 

16' 

18' 

Il8i.3         | 

2314.0 

120.5 

i87 

20' 

1183.0      g 

2317.3 

120  .  9 

20' 

22' 

1184.8      < 

2320  .6 

121  .  2 

22' 

24' 

1186.5 

2323-8 

121  .6 

24' 

26' 

1188.3 

2327.1 

I2I-9 

26' 

28' 

1190  .0 

2330.4         122.3 

28' 

60 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

23°  30' 

1191.7 

2333-6 

122  .6 

23°  30' 

32' 

II93-5                    2336.9 

123.0 

32; 

34' 

1195.2                    2340.2 

123-3 

34 

36' 

1197.0 

2343-5 

123-7 

36' 

38' 

1198.7 

2346.7 

I24.O 

38' 

40' 

I2OO  .4 

2349.9 

124.4 

40' 

42' 

I2O2  .  2 

2353.2 

124.7 

42' 

44' 

1203.9 

2356.4 

125.0 

44' 

46' 

1205.7 

2359-7 

125.4 

46' 

48' 

1207.4 

2363-0 

125.8 

48' 

50' 

1209.1          4 

2366  .  2 

126  .  2 

50' 

52' 

1210.9 

2369  .  5 

126.5 

52' 

54' 

1212  .6           »§ 

2372.7 

126  .9 

54' 

56' 

I2I4.4           -g 

2376.0 

127  .2 

56' 

58' 

I2I6.I            •£ 

2379.2 

127  .6 

58' 

24° 

03 
1217.9            fc 

2382.5 

128.0 

24° 

2' 

1219.6          £ 

2385-8 

128.3 

2' 

4' 

I22I.4            N' 

2389.1 

128.7 

4' 

6' 

I223.I    ^«£ 

2392.3 

129  .O    ^G"'* 

6' 

8' 

1224.9  j°:  T3 

2395-6 

129  .4    6-    - 

8' 

10' 

1226  .6  "g 

2398.8 

129.8  J 

10' 

12' 

1228.3  -a3 

2402  .  I 

130.1  .£:  : 

12' 

14' 

I230.I     u 

2405.3 

130.5^ 

14' 

16' 

1231.8-2^ 

2408  .6 

130.8^  = 

16' 

18' 

1233-5   OM4 

2411  .8 

131  *2  1;  25 

18' 

20' 

I235  -3  "  "*6 

2415.1 

131  .6  „_     M 

20' 

22' 

1237-°  I5  £ 

2418.4 

132.05:  - 

22' 

24'  |     1238.8^     2 

2421  .6 

132.3    ' 

24' 

26' 

1240.5        £ 

2424.9 

132,7 

26' 

28' 

1242.2         g 

2428  .  i 

J33-  I 

28' 

30' 

1244.0        «r 

2431.4 

133-5 

30' 

32' 

1245.7         o 

2434-7 

133-8 

32' 

34' 

1247.5        -a 

2437-9 

134-2 

34'. 

36' 

1249-3        3 

2441.2 

134.6 

36; 

38' 

1251  .0 

2444.4 

i35-o 

38' 

4°' 

1252.7 

2447.7 

135-4 

40' 

42' 

I254-5 

2451.0 

135-8 

42' 

44' 

1256.3 

2454.2 

136.2 

44' 

46' 

1258.0 

2457-5 

136-5 

46' 

48' 

1259-7 

2460  .  8 

136.9 

48' 

5°' 

1261  .4 

2464  .0 

137.2 

50/ 

52' 

1263  •  2 

2467.3 

137-6 

52/, 

54' 

1265  .O 

2470.5 

138.0 

54 

56' 

1266  .  7 

2473.8 

138.3 

56 

58' 

1268.5 

2477.0 

138.7 

58' 

61 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

25° 

1270  .  2 

2480  .  2 

I39-1 

25° 

2f 

1272  .0 

2483-5 

J39-5 

2' 

4' 

1273-7 

2486.7 

139-9 

4' 

6' 

I275-5 

2490  .0 

140.2 

6' 

8' 

1277.2 

2493.2 

140  .6 

8' 

10' 

1278.9 

2496.4 

141  .0 

10' 

12' 

1280  .  7 

2499.7 

141  .4 

12' 

14' 

1282  .4 

2502.9 

141  .8 

14' 

16' 

1284.2 

2506  .  2 

142  .  2 

16' 

18' 

1286.0 

2509-5 

142  .6 

18' 

20' 

1287  .  7  10  o4 

2512.7 

142  .9 

20' 

22' 

I289-5  c-  . 

2516.0 

143-3 

22' 

24' 

1291  .  2  X"    " 

2519.2 

24' 

26' 

I293.0rt 

2522.5 

144.0 

26' 

28' 

1294.8  £== 

144.4 

28' 

3o' 

1296  .  5  £.  , 

2529  .O 

144.8 

3O/ 

32' 

I298.3^Voo 

2532.3 

145.2 

32' 

34' 

1300.0       0     M     CJ 

2535-5 

145.6 

34' 

36' 

1301.8^^ 

2538.8 

146  .0  ">  <*;* 

36' 

38' 

2542.0 

146.4  6.  . 

38' 

40' 

1305.2 

2545-2 

146.8^ 

40' 

42' 

1307.0 

2548.5 

147-2-g:  : 

42' 

44' 

1308.7 

255J-7 

147  .6  <n 

44' 

46' 

1310  .  5 

2555-o 

148.0,0:  : 

46' 

48' 

1312.2 

2558-2 

148    .4    0    TjT- 

48' 

CS    t^O 

5°' 

1314.0 

2561.4 

148.8^  - 

50' 

52', 

1315.7 

2564-7 

149.2  ^:  : 

52' 

54' 

1317  .  5 

2567.9 

149.6  " 

54' 

56' 

1319.2 

2571.2 

150.0 

56' 

58' 

1321.0 

2574-4 

150.4 

58' 

2«° 

1322.8 

2577-7 

I5°«7 

26° 

2' 

1324.5  >Ao4 

2581  .0 

151.1 

2' 

4' 

1326.3 

2584.2 

I5I-5 

4f 

6' 

1328.1  55-  = 

2587-5 

i5i-9 

6' 

8' 

1329.8-3 

2590.7 

152.3 

8' 

10' 

I33I  -5  £:  : 

2593-9 

152.7 

10' 

12' 

J333-2  g.  , 

2597.2 

i53.i 

12' 

14' 

i335-o-"0^ 

2600  .4 

J53-5 

14' 

16' 

1336.7  6*? 

2603.7 

16' 

18' 

1338.  5|3£ 

2607  .0 

154.3 

18' 

20' 

1340.3  |:  : 

26lO  .  2 

154.7 

20' 

22' 

1342.0     ' 

2613.5 

22' 

24' 

1343-8 

26l6.8 

J55-5 

24' 

26' 

1345-6 

262O  .O 

155  .9 

26' 

28' 

1347-4 

2623  .  2 

156.3 

28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURV 


'E. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

26°  30' 

I349-1 

2626  .4 

156.7 

26°  30' 

32' 

I350-9 

2629.7 

I57-I 

32' 

34' 

l352-7 

2632.9 

157-5 

34' 

36' 

1354-4 

2636  .  I 

157-9 

36' 

38' 

KSS6-2 

2639.4 

158.3 

38' 

40' 

1357-9 

2642  .6 

158.7 

40' 

42' 

1359-7 

2645.9 

I59-1 

42' 

44' 

!36i.5 

2649.  J 

159-5 

44' 

46' 

1363.2 

2652.4 

160  .0 

46' 

48' 

1365-0 

2655.6 

160  .4 

48' 

50/ 

1366.7         4 

2658.9 

160.8 

50' 

52' 

1368.5 

2662  .2 

161  .  2 

52' 

54' 

1370-2       £ 

2665  .4 

161.6 

54' 

56' 

i372.o        -3 

2668.7 

162  .0 

56' 

58' 

1373-8        £ 

2671.9 

162  .4 

58' 

27° 

K) 

1375-5         * 

2675.1 

162.8 

27° 

2' 

1377-3        * 

2678.4 

163.2 

2' 

4' 

1379.0         * 

2681.6 

I63.6^4 

4' 

6' 

1380.8  «AdR 

2684.9 

164.0 

6' 

8' 

1382.6  c:  ;§ 

2688.1 

164.5!-  : 

8' 

10' 

1384.4^    ^ 

2691  .3 

164.  9g 

10' 

12' 

1386.2-^ 

2694  .6 

165.  3  -a'  : 

12' 

14' 

1388.0^ 

2697.8 

165.7^. 

14' 

16' 

13*6*7-* 

2701  .  i 

166.1  «£-  - 

1  6' 

18' 

i39i.5]^4 

2704.3 

166.5--  25 

18' 

20' 

1393-2  j^d 

2707.5 

166  .9  ^d 

20' 

22' 

1395-03:  Jz: 

2710  .8 

167.  4<~  : 

22' 

24' 

1396.7^     g 

2714.0 

167.8 

24' 

26' 

J398-5        i| 

2717.2 

168.2 

26' 

28' 

1400.3         ^ 

2720.4 

168.6 

28' 

30' 

1402.0        ^ 

2723.6 

169  .0 

30' 

32' 

1403.8        ? 

2726.8 

169.4 

32' 

34' 

J405.6        ^ 

2730.0 

169.8 

34' 

36' 

I4°7-3        ^ 

2733-3 

170.3 

36' 

38' 

1409.  i 

2736.5 

170.7 

38' 

40' 

1410.8 

2739.8 

171.1 

40' 

42' 

1412  .6 

2743.0 

171.6 

42' 

44' 

1414.4 

2746.3 

172  .0 

44' 

46' 

1416  .  i 

2749-5 

172.4 

46' 

48' 

1417.8 

2752.7 

172.9 

48' 

5°: 

1419.6 

2756.0 

!73-3 

5°' 

5* 

1421.4 

2759.2 

173.7 

52' 

54 

1423.1 

2762  .5 

174.2 

54' 

56' 

1424.9 

2765.8 

174.6 

56' 

58' 

1426.7 

2769  .0 

175-0 

58' 

63 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

28° 

1428.5 

2772.2 

J75-4 

28° 

2f 

I430-3 

2775-5 

175-8 

2' 

4' 

1432.0 

2778.7 

176.3 

4' 

6' 

1433-8 

2782  .0 

176.7 

6' 

8' 

1435-6 

2785-2 

177.2 

8' 

10' 

1437-4 

2788.4 

177.6 

10' 

12' 

1439.2 

2791.7 

178.0 

12' 

14' 

1441  .0 

2794.9 

178.5 

14' 

16' 

1442.7 

2798.1 

178.9 

16' 

18' 

1444-5 

2801.3 

179-3 

jSf 

20' 

1446.2          4 

2804.  5 

179.7 

.     20' 

22' 

1448.0          ^ 

2807.8 

l8o.2 

22' 

24' 

1449.7         ^ 

2811  .0 

180.6 

24' 

26' 

i45J-5        "g 

2814.  2 

iSl.O 

26' 

28' 

1453-3        £ 

2817.5 

181.5 

28' 

3o' 

I455-1        | 

2820  .  7 

181  .9 

30' 

32' 

1456.9 

2824  .0 

182.3 

;t  32' 

34' 

1458.7     -    -o 

2827  .  2 

182.8    .  .  . 

34' 

36' 

1460.5  ;?*£ 

2830.5 

183.2  $$t 

36' 

38' 

1462.  3£-  ^ 

2833-7 

183-7^:  : 

38' 

40' 

1464.0  £ 

2836.9 

184.1  -^ 

4o' 

42' 

1465-  8  £: 

2840  .  2 

184.6-^  : 

42' 

44' 

1467.6  g. 

2843-4 

185-0^ 

447 

46' 

1469.3  ooo 

2846.6 

185.  4£:  : 

46' 

48' 

I471-*  °£4 

2849.8 

l85-9^22 

48' 

5o' 

1472.9^** 

2853.0 

186.3^ 

50' 

52' 

1474.  7  3:  ,§ 

2856.3 

186.7^ 

-          52' 

54' 

i476-5        -5 

2859.5 

187.2 

54' 

S^' 

'478-3        | 

2862.7 

187.6 

56 

58' 

1480.0        co 

2866.0 

188.1 

58' 

29° 

1481.8        1 

2869.2 

188.5 

29° 

a' 

1483-6         ^ 

2872.5 

189  .0 

2' 

4' 

1485-4     a 

2875-7 

189.4 

4' 

6' 

1487.1     ^ 

2878.9 

189.9 

6' 

8' 

1488.9     < 

2882.1 

190.3 

8' 

10' 

1490.7 

2885.3 

190.7 

10' 

12' 

1492.5 

2888.5 

191  .2 

12' 

14' 

1494.3 

2891.8 

191  .6 

14' 

16' 

1496  .0 

2895  .O 

192  .  i 

16' 

18' 

1497.8 

2898.2 

192.5 

18' 

20' 

1499.6 

2901  .4 

193.0 

20' 

22' 

1501  .4 

2904  .  6 

193-5 

22' 

24' 

1503-2 

2907.9 

193-9 

24' 

26' 

1505-0 

2911  .  i 

194.4 

26' 

28' 

1506.7 

2914.3 

194.8 

28' 

64 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

29°  30' 

1508.5 

2917.5 

195-3 

29°  30' 

32' 

I5I0.3 

2920  .8 

195.7 

32; 

34' 

1512  .0 

2924.0 

196.2 

34 

36' 

I5I3.8    .   . 

2927.2 

196  .6 

36 

38' 

1515-6"** 

2930.5 

197.1 

38' 

40' 

1517.4^=  = 

2933.7 

197-5 

40' 

42' 

1519.2-3 

2937.0 

198  .0 

42' 

44' 

1521.0-*:  : 

2940  .  2 

198.4 

44' 

46' 

1522  .  7  ^ 

2943.4 

198.9 

46' 

48' 

1524.5  £=  = 

2946  .6 

199.3 

48' 

O  GO     Tf 

50' 

1526.3     C    i-'    rr> 

2949.8 

199.8 

5° 

52' 

1528.0  «'^R 

2953.0 

200.2 

52^ 

54' 

1529.8?,  : 

2956.3 

200.7 

54 

56' 

1531  .6  "^ 

2959.5 

2OI  .  I 

56 

58' 

1533.4 

2962  .7 

2OI  .6 

58' 

30° 

1535.2 

2965.9 

2O2  .  I 

30° 

2' 

1537.0 

2969.1 

2O2  .6 

2' 

4' 

1538.8 

2972.3 

203.0          .    . 

4' 

6' 

1540.6        ^_ 

2975.6 

203.5    ^^^ 

6' 

8' 

1542.4 

2978.8 

203.  9^3  : 

8' 

10' 

0 

1544.2      z 

2982  .0 

204.4^3 

10' 

12' 

1546.0      -g 

2985.3 

204.9  :§r  : 

12' 

14' 

1547-8      *a 

2988.5 

205.3^ 

14' 

16' 

1549.6      t£ 

2991.7 

205  .8  £z  : 

16' 

18' 

1551.4     £ 

2994.9 

206.3  "3  itv? 

18' 

20' 

1553.2  t^s 

2998.  i 

206.8,3     w 

20' 

22' 

1554.9,6-  £ 

3001.3 

207.2  5j:  = 

22' 

34' 

1556.7^  ^ 

3004.5 

207.7 

24' 

26' 

3007.7 

208.1 

26' 

28' 

1560.  3  1" 

3010.9 

208.6 

28' 

30' 

1562  .1  jr: 

3014.1 

2t>9.1 

3°; 

32' 

1563  ,9  ^09  . 

3017-3 

209.5 

32 

3< 

*565-7  §5- 

3020.5 

2IO  .O 

zi 

36' 

1567-4^  §, 

3023.8 

2IO  .5 

36 

38' 

1569.23=5 

3027.0 

211  .O 

38' 

40' 

2 
1571.0       •£ 

3030.2 

2II.5 

40; 

42' 

1572.8       ™ 

3033.4 

212  .0 

42' 

44' 

1574.6       £ 

3036.6 

212  .4 

44' 
sf 

46' 

1576.4       ^ 

3039.9 

212  .9 

46' 

48' 

1578.2  ^       c 

3043.1 

213.4 

48' 

50', 

1580.0       5 

3046.3 

213.9 

5°' 

52' 

1581.8      < 

3049.5 

214.4 

52' 

54' 

1583.6 

3052.8 

214.8 

54' 

56' 

1585.4 

3056.0 

215.3 

56 

58' 

1587.2 

3059-2 

215.8 

58 

65 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 

Angif. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Agnle. 

31° 

1588.9 

3062  .4 

216.3 

31° 

2' 

1590.7 

3065.6 

216.8 

2' 

4' 

1592.5 

3068.8 

217.3 

4' 

6' 

1594.3 

3072.0 

217.7 

6' 

8' 

1596.1 

3075-2 

218.2 

8* 

10' 

1597-9 

3078.4 

218.7 

10' 

12' 

1599-7 

3081.6 

219  .  I 

I27 

14' 

1601  .  5 

3084.8 

219  .6 

14' 

16' 

1603.3 

3088.0 

220  .  I 

16' 

18' 

1605.1 

309L3 

22O  .  6 

18' 

20' 

1606  .9 

3094.5 

221  .  I 

20' 

22' 

1608.7 

3097.7 

221.6 

22X 

24' 

1610.5 

3100  .9 

222  .  I 

24' 

26' 

1612.3 

3104.1 

222  .  5 

26' 

28' 

1614  .  i 

3I07-3 

223.0 

28' 

30' 

1615.9 

3110.5 

223.5 

3°' 

32' 

1617.7 

3113-7 

224  .O 

32' 

34 

l6*9-5  ^4 

3116.9 

224.5  ^4 

34' 

36 

1621.3 

3120.1 

225  .0 

36' 

38' 

1623.1  jO:  : 

3123.3 

225.5  £  =  - 

38' 

40' 

1624.9  13 

3126.5 

226  .0  15    - 

40' 

42' 

1626  .  7  'Br  : 

3129.7 

226.5  '&: 

42' 

44' 

1628.  5^  m 

3132.9 

227.0^ 

44' 

46' 

1630.3  £~  : 

3136.2 

227.5  £"  " 

46' 

48' 

1632.1  £J£ 

vo  v,  O 

3139-4 

228.0^! 

48' 

50/ 

1633.9^*- 

3142.6 

228.4^ 

50' 

52/, 

1635  •  7  ^  : 

3I45-8 

228.  9^:  : 

52' 

C  A 

1637-5 

3149.0 

229.4 

54' 

56' 

1639-3 

3i52-2 

229.9 

56; 

58' 

1641  .1 

3I55-4 

230.4 

32° 

1642  .9 

3158.6 

230.9 

32° 

2f 

1644.7 

3161.8 

231.4 

2f 

4' 

1646.5 

3165.0 

231.9 

4' 

6' 

1^8-8.3 

3168.2 

232.4 

6' 

8' 

1650.  i 

3i7i-4 

332  .9 

8' 

10' 

1651.9 

3174.6 

233-4 

10' 

12' 

1653.7 

3177.8 

233-9 

I27 

14' 

l655-5 

3181  .0 

234-4 

14' 

16' 

1657-3 

3184.2 

234.9 

16' 

18' 

1659.1 

3187.4 

235-4 

18' 

20' 

1660.  9 

3190.6 

235.9 

20' 

22' 

1662  .  7 

3193.8 

236.4 

22' 

24' 

1664.5 

236.9 

24' 

26' 

1666.3 

3200.  2 

237.4 

26' 

28' 

1668.1 

3203.4 

237-9 

28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

32°  30' 

1669.9 

3206  .  6 

238.4 

32°  30' 

3*' 

1671.7 

3209.8 

238.9 

32' 

34' 

1673.5 

3213.0 

239-4 

34' 

36' 

1675.3 

3216.2 

•     239.9 

36' 

38' 

1677.1 

3219-4 

240.5 

38' 

40' 

1679  .O 

3222  .6 

241  .O 

40' 

4*' 

1680.8 

3225.8 

24L5 

42' 

44' 

1682.6 

3229.0 

242  .0 

44' 

46' 

1684.4 

3232.2 

242.5 

46' 

48' 

1686.2 

3235.4 

243-0 

48' 

50/ 

1688.1 

3238-6 

243-5 

5o; 

52' 

1689.9 

3241.8 

244.0 

54' 

1691.7 

3245-0 

244.6 

54' 

56' 

1693-5 

3248.2 

245-1 

56' 

58' 

1695.3 

325L4 

245-6 

58' 

33° 

1697.2 

3254.6 

246.1 

33° 

2f 

1699.0 

3257.8 

246.6 

2' 

4' 

1700.8 

3261  .0 

247  .2      ... 

4' 

6' 

1702  .7  ^  ^^ 

3264.2 

247-7    ""*» 

6' 

8' 

1704.5  ;     - 

3267.4 

248.2  6,  : 

8X 

10' 

1706.  3  £~  : 

3270.6 

248.  7  -a 

10' 

12' 

1708.1  *§ 

3273.8 

249.2-5=  = 

12' 

14' 

1709.  9  'ST  : 

3277.0 

249.7^ 

14' 

16' 

C/2 

1711.7 

3280.2 

250.  3£:  : 

1  6' 

18' 

I7I3-5  ^_:0: 

3283.4 

250.8  £££ 

1  8' 

20' 

1715-3  ££j 

3286.6 

251-3-0 

20' 

22' 

1717.1  ***- 

3289.8 

251.83=  = 

22' 

24' 

1718.9  *c:  : 

3293.0 

252.3 

24' 

26' 

1720.8  Y 

3296.2 

252.8 

26' 

28' 

1722  .6 

3299.3 

253-3 

28' 

30/ 

1724.4 

3302.5 

253-9 

30' 

32' 

1726.2 

3305.7 

254-4 

32 

34' 

1728.0 

3308.9 

255.0 

34 

36' 

1729.9 

3312  .  i 

255.5 

36 

38' 

I73L7 

3315.3 

256.0 

38' 

40' 

1733.5 

3318.5 

256-5 

40' 

42' 

1735.3 

3321.7 

257.0 

42' 

44' 

1737.2 

3324.9 

257.5 

44' 

46' 

3328.1 

258.0 

46' 

48' 

1740.8 

3331-2 

258.5 

48' 

50' 

1742.6 

3334-4 

259.1 

5°' 

I*' 

1744.4 

3337-6 

259.6 

52', 

54'. 

1746.2 

3340.8 

260.2 

54' 

56' 

1748  .0 

3344-0 

260.7 

56' 

58' 

1749.9 

3347-2 

261  .3 

58' 

67 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

34° 

I75L7 

3350.4 

261.8 

34° 

2' 

1753.6 

3353-6 

262  .4 

2' 

4' 

1755.4 

3356.8 

262  .9 

4' 

6' 

1757.2           4 

3360.0 

263.5 

6' 

8' 

1759-0          ^ 

3363.1 

264.0 

8' 

10' 

1760.8         g 

3366.3 

264.5 

10' 

12' 

1762.7         "g 

3369.5 

265  .  I 

12' 

14' 

1764.5      £ 

3372.7 

265.6 

14' 

16' 

1766.3       £ 

3375-9 

266  .2 

16' 

18' 

1768.1       ^ 

3379-0 

266.7 

18' 

20' 

1769.9       o 

3382.2 

267.2 

20' 

22' 

i77i-7       £ 

267.8 

22' 

24' 

1773.6      ^ 

3388^6 

268.3 

24' 

26' 

1775-4 

3391-8 

268.9 

26' 

28' 

1777.2 

3395-0 

269.4 

28' 

30' 

1779.0 

3398.1 

269.9 

30' 

32' 

1780.9 

3401.3 

270.5 

32' 

34' 

1782.7  ^4 

3404.5 

270.0 

34' 

36' 

1784.5 

3407-7 

271  .6  iA»4 

36' 

38' 

I786.3|=    = 

3410.9 

272.1  ^  _H 

38' 

40' 

1788.2  *§ 

3414.0 

272  .6  2 

40' 

42' 

1790.  I  £:: 

3417.2 

273.2  .§:  : 

42' 

44' 

I79L9    g-    - 

3420.4 

273-7  & 

44' 

46' 

3423.5 

274.3  gs  r 

46' 

48' 

1795^12$ 

3426.7 

274.8^o  H 

48' 

5°; 

1797.4^" 

3429.9 

275.3  °"^ 

5°; 

52' 

1799.2^:  : 

3433-1 

275.  9S:  : 

52 

54' 

1801  .  i 

3436.3 

276.  4< 

54' 

56' 

1803.0 

3439.4 

277.0 

56' 

58' 

1804.8 

3442.6 

277.5 

58' 

,35° 

1806.  6 

3445  -  8 

278.1 

35° 

2' 

1808.5         4 

3449-0 

278.6 

2' 

4' 

1810.3 

3452.1 

279.2 

4' 

6' 

1812  .  i        £ 

3455-3 

279.7 

6' 

8' 

1813-9        -g 

3458.5 

280.2 

8' 

10' 

1815.7       £ 

346i.7 

280.8 

10' 

12' 

1817.6        g 

3464.9 

281.3 

12' 

14' 

1819.4       *£ 

3468.0 

281.9 

14' 

16' 

1821.3        * 

3471-2 

282.4 

16' 

18' 

1823.1 

3474-4 

283.0 

18' 

20' 

1824.9     3 

3477-6 

283.6 

20' 

22' 

1826.8 

3480.8 

284.1 

22r 

24' 

1828.6 

3484.0 

284.7 

24' 

26' 

1830.4 

3487.1 

285.2 

26' 

28' 

1832.3 

3490.3 

285.8 

28' 

68 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle.     . 

35°  30' 

1834.1 

3493-5 

286.4 

35°  30' 

3*' 

1836.0 

3496.6 

286.9 

32' 

34' 

I837-8          4 

3499-8 

287.5 

34' 

36' 

1839.7 

3503-0 

288.0 

36' 

38' 

1841-5         & 

3506.2 

288.6 

38' 

40' 

1843-3     'S 

3509.3 

289.2 

40' 

42' 

1845-2      £ 

35*2.5 

289.7 

42' 

44' 

1847.0        1* 

290.3 

44' 

46' 

1848.9     § 

35i8l8 

290.8 

46' 

48' 

1850.7          4 

3522.0 

291.4 

48' 

5°' 

1852.5         £ 

3525-2 

292  .0 

50' 

5*' 

1854.3     3 

3528.4 

292  .6 

52' 

54' 

1856.2 

3531  .  5 

293.1 

54' 

56' 

1858,0 

3534-7 

293-7 

56' 

58' 

1859.9 

3537-9 

294-3 

58' 

36° 

1861.7 

354i.i 

294.9 

36° 

2' 

1863.6 

3544-3 

295-5 

2' 

4' 

1865.4 

3547-4 

296.0 

4' 

6' 

1867.3                      " 

3550-6 

296.6  ^<>j 

6' 

8' 

1869.2                    g 

3553-8 

297  .  1   o-  . 

8' 

10' 

1870.9        13 

3557-0 

297-7^  " 

10' 

12' 

1872.8      •£ 

356o.2 

298.3  .h-  - 

12' 

14' 

1874.6      & 

3563-3 

298.  8  c&"  " 

14' 

16' 

1876.4      £ 

3566.5 

299.  4J5:  : 

1  6' 

18' 

1878.3     5 

3569-6 

3OO.O      J>0    H 

1  8' 

20' 

1880.  i  "»£ 

3572.8 

3°°   '6* 

20' 

22' 

1882.  o£:  ? 

3576.0 

301  .2  •£::  : 

22' 

24' 

1883.8*  < 

3579-1 

3°I-7< 

24' 

26' 

1885.74= 

3582.3 

302.3 

26' 

28' 

.i887.5£ 

3585.5 

302.9 

28' 

3°; 

1889.  3  Is 

3588.6 

303.5 

30' 

1891  .  2  ^  "• 

3591.8 

304.0 

32 

!*' 

1893.0    £$? 

3594-9 

304.6 

34 

36' 

1894.85,    6 

3598.1 

305.2 

36 

38' 

1896.7  <"  2 

3601.3 

305-8 

38' 

40' 

1898.5    4 

3604.4 

306.4 

40' 

42/ 

1900.4    ^ 

3607.6 

307.0 

42' 

44' 

1902.2          £ 

3610.7 

307.5 

44' 

46' 

1904.1        IJ? 

3613-9 

308.1 

4| 

48' 

1905.9           o 

3617.1 

308.7 

48' 

5°' 

1907.8      § 

3620.2 

309.3 

5°' 

S^' 

1909.6        < 

3623.4 

309.9 

52/ 

54' 

1911.5 

3626.5 

310.4 

5i 

56' 

1913.3 

3629.7 

311.0 

56' 

58' 

1915.2 

3632  .8 

3H.6                              58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

37° 

1917.1 

3636.0 

312.2 

37° 

a' 

1919  .O 

3639  -1 

312.8 

2' 

4' 

1920  .8 

3642.3 

313.4 

4' 

6' 

1922.7         ? 

3645.5 

314.0 

6' 

8' 

i  9&4  .  5         6 

3648.6 

314.6 

8' 

10' 

1926.4        -j 

3651-8 

3I5.2 

10' 

12' 

1928.3        £ 

3654.9 

315.8 

12' 

14' 

1930.1         tw 

3658.1 

316.4 

14' 

16' 

1932.0         £ 

3661.3 

3I7-0 

16' 

18' 

1933.8         « 

3664.5 

317.6 

18' 

20' 

1935-7  ^£ 

3667.6 

318.1 

20< 

22' 

1937.6  °:  3 

3670.8 

318.7 

22' 

24' 

1939-4-    < 

3673.9 

3*9-3 

24' 

26' 

1941  .3  & 

3677.1 

3J9-9 

26' 

28' 

1943.1  co 

3680  .  2 

320.5 

28' 

30' 

1945.0  J: 

3683.3 

321.1 

3o' 

32' 

1946.9124 

3686.5 

321.7 

32' 

34' 

1948.7  £3H. 

3689.6 

322.3 

34' 

36' 

1950.6^  £ 

3692.8 

322.9  «A  »  j 

36' 

38' 

1952.  4<f  -3 

3695-9 

323-5   d.  : 

38' 

40' 

1954-3        I 

3699.1 

324-  I  5" 

40' 

42' 

1956.2         £ 

3702.2 

324.  7  1:: 

42' 

44' 

1958.0        ^ 

3705.5 

325.3^ 

44' 

46' 

1959-9         4 

3708.6 

325.9,0:  = 

46' 

48' 

1961.7     a 

37II.8 

326.5    r*\o  N 

48' 

50' 

1963.6     | 

37M.9 

327.I    """ 

50' 

52' 

1965-5 

3718.0 

327.7^:  : 

52' 

54' 

1967.3 

3721.2 

328.  3^ 

54' 

56' 

1969  .  2 

3724.5 

328.9 

56' 

58' 

I97I.O 

3727.6 

329-5 

58' 

38° 

1972.9 

3730.7 

330.2 

38° 

2' 

1974.8 

3733-8 

330.8 

2' 

4' 

1976.6    ^^^ 

3737-0 

331-4 

4' 

6' 

1978.5           > 

3740.1 

332.o 

6' 

8' 

1980.3,0:  : 

3743.3 

332.6 

8' 

10' 

1982  .  2  'd 

3746.4 

333-2 

10' 

12' 

1984.0  -g,"    : 

3749-5 

333.8 

12' 

14' 

1985.9^ 

3752.7 

334.5 

14' 

16' 

1987-  7«2:  : 

3755-8 

335-1 

16' 

18' 

1989-6  ?n 

3759-o 

335-7 

18' 

20' 

to  u-j  O 
I99I.5    cj^t^ 

3762.2 

336.3 

20' 

22' 

1993  -4  ?:  : 

3765.3 

336.9 

22' 

24' 

1995.  3^ 

3768.5 

337-5 

24' 

26' 

1997.1 

3771.6 

338.i 

26' 

28' 

1999.0 

3774-8 

338.7 

28' 

70 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

38°  30' 

2OOO  .8 

3777-9 

339-3 

38°  30' 

32' 

2OO2  .  7 

3781.1 

339-9 

32' 

34' 

2004.5 

3784.2 

340.6 

34' 

36' 

2006.4          4 

3787-4 

341.2 

36' 

38' 

2008  .3 

3790.5 

341.8 

38' 

40' 

2OIO  .  2            £ 

3793-7 

342.4 

40' 

42' 

2012.  I            *g 

3796.8 

343-o 

42' 

44' 

2014.0           £ 

3800  .0 

343-7 

44' 

46' 

2015.8        C 

3803.1 

344-3 

46' 

48' 

2017.7     - 

3806.2 

344-9 

48' 

50' 

2019.6         £" 

3809.4 

345-5 

50' 

5*' 

2021.5     £ 

3812.5 

346.1 

52' 

54' 

2023.4     5 

3815-7 

346.8 

54' 

56' 

2025.3 

3818.9 

347-4 

56' 

58' 

2027  .2 

3822.0 

348.0 

58' 

39° 

2O29  .O 

3825.1 

348.6 

39° 

2' 

2030.9 

3828.2 

349-3 

2' 

4' 

2032  .8    ... 

3831.4 

349-9  ^  .  . 

4' 

6' 

2034.    7'    ^H 

3834-5 

350.5""? 

6' 

8' 

2036.6  ,o:  : 

3837-7 

351-2   c,  5 

8' 

10' 

2038.4-3 

3840.8 

35i.  8-g 

10' 

12' 

2040.3  -a:  : 

3843-9 

352.  4  -a:  = 

12' 

14' 

2042.2^ 

3847-1 

353.1^ 

14' 

16' 

2044.  i  £:  : 

3850.2 

353-  7-:  : 

16' 

18' 

2046.  o||| 

3853.4 

354.3  ^2 

18' 

20' 

2047  -8  N  ^^ 

3856.5 

354-  9-d 

20' 

22' 

2049.7  S:  : 

3859-6 

355-  5<:  : 

22' 

24' 

2051  .6  £1 

3862.8 

356.2 

24' 

26' 

2053-5 

3865-9 

356.8 

26' 

28' 

2055.4 

3869.1 

357-5 

28' 

30' 

2057.2 

3872.2 

358.1 

30' 

32' 

2059.1          4 

3875-3 

358.7 

32' 

34' 

2061  .0 

3878.5 

359-4 

34 

36' 

2062.9         j§ 

3881.6 

360.0 

36' 

38' 

2064.8         13 

3884-7 

360.7 

38' 

40' 

2066.6         '& 

3887.9 

361.3 

40' 

.     42' 

2068.5          £ 

3891.0 

362.0 

42' 

44' 

2070.4        ^ 

3894.2 

362.6 

44' 

46' 

2072.3          £ 

3897-3 

363-3 

46' 

48' 

2074.2          *- 

3900.4 

363-9 

48' 

50' 

2076.0         ^ 

3903.5 

364-5 

50' 

52' 

2077.9 

3906.6 

365-2 

52 

54' 

2079.8 

3909.8 

365-8 

54 

56' 

2081  .  7 

3912.9 

366.5 

56 

58' 

2083.6 

3916.0 

367.1 

=;8' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

4O° 

2085  .  4 

3919.2 

367.7 

4O° 

2' 

2087.3 

3922.3 

368.4 

2' 

4' 

2089  .  2 

369.0 

4? 

6' 

2091  .  I 

3928.6 

369.7 

6' 

8' 

2093.0 

3931-7 

370.3 

8' 

10' 

2094.9 

3934-8 

371.0 

10' 

12' 

2096.8 

3937-9 

371-6 

12' 

14' 

2098.7 

3941.0 

372.3 

14' 

16' 

2100.6 

3944-2 

372.9 

1  6' 

18' 

2IO2  .5 

3947-3 

373.6 

18' 

20' 

2104.3             £ 

3950-5 

374-2 

20'. 

22' 

2IO6.  2            *?< 

3953.6 

374-9 

22' 

24' 

2I08.I            "g 

3956.8 

375-5 

24' 

26' 

2IIO.O           •£ 

3959-9 

376.2 

26' 

28' 

2111.9        «J 

3963.0 

376.8 

28' 

3°' 

2H3.7           ^ 

3966.1 

377-5 

30' 

32' 

2II5.6             ^ 

3969.2 

378.1 

32' 

34' 

2117.5    l?°>° 

3972.4 

378.8    ... 

34' 

36' 

2119.4,°:  .3 

3975.5 

379.4100? 

36' 

38' 

2121.3^' 

3978.6 

380.1   c,.  : 

38' 

40' 

2123.  2  1,: 

3981  .8 

380.8^ 

40' 

42' 

2125.1?  . 

3984.9 

381.  4  •&-  = 

42' 

44' 

2127  .0  <~V 

3988.1 

382.  !<« 

44' 

46' 

2128.9    o<?    • 

3991.2 

382.7^    = 

46' 

48' 

2130.8     £$« 

3994.3 

383.4   ^^^ 

48' 

50' 

2132.  7  I5  1 

3997-4 

384.  i  ^  ; 

50' 

52' 

2134.  6^    15 

4000.5 

52' 

54' 

2136.5      -a 

4003.7 

385^ 

54' 

56' 

2138.4      ? 

4006  .8 

386.0 

56' 

58' 

2140.3      £ 

4009  .  9 

386.7 

58' 

41° 

2142.2      ^ 

4013.1 

387.4 

41° 

2 

2144.1      £• 

4Ol6  .  2 

388.1 

2' 

4' 

2146.0      i§ 

4019.3 

388.7 

4' 

6' 

2147.9      < 

4022.5 

389.4 

6' 

8' 

2149.8 

4025.6 

390.0 

8' 

10' 

2151.7 

4028.  7 

390.7 

10' 

12' 

2153.6 

4031  .8 

391.4 

12' 

14' 

2155.5 

4034.9 

392.0 

14' 

16' 

2157.4 

4038.0 

392.7 

1  6' 

18' 

2159-3 

4041  .  2 

393.4 

18' 

20' 

2l6l  .2 

4044-3 

394.1 

20' 

22' 

2I63.I 

4047-4 

394.8 

22' 

24' 

2165  .O 

4050.5 

395-4 

24' 

26' 

2166  .  9 

4053.6 

396.1 

26' 

28' 

2168.8 

4056.7 

306.7 

28' 

IX.— FUNCTIONS  OF  A^  ONE-DEGREE  CURVE. 


Whole 

Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

41°  30' 

2170.7 

4059.8 

397-4 

41°  3°' 

32' 

2172.6 

4062  .  9 

398.1 

32' 

34' 

2174.5 

4066  .0 

398.8 

34' 

36; 

2176.4 

4069  .  2 

399-5 

36' 

2178.3 

4072.3 

400.2 

38' 

40' 

2180.3 

4075-4 

400.9 

40' 

42' 

2l82  .  2 

4078.5 

401  .6 

42' 

44' 

2184.  I 

4081  .6 

4O2  .  2 

44' 

46' 

2186.  o 

4084.7 

402.9 

46' 

48' 

2188.0 

4087.9 

403.6 

48' 

50/ 

2189.9          4 

4091  .0 

404.3 

5°' 

52/ 

2191.8         " 

4094.1 

405.0 

52' 

54' 

2193-7         * 

4097.2 

405.6 

54' 

56' 

2195-6         -g 

4100.3 

406.3 

56' 

58' 

2197-5            £ 

4103.4 

407.0 

58' 

42° 

2199.4     "£ 

4106.6 

407.7 

42° 

2' 

2201.3            ^ 

4109.7 

408  .4 

2' 

4' 

2203.3             £ 

4112.8 

409.0 

4' 

6' 

2205.2  •«?«>*- 

4115.9 

409.7  ^i 

6' 

8' 

2207  .  1  jg:  Ig 

4119.0 

410.4^-  . 

8' 

10' 

22O9  .O  'rt 

4122  .  i 

4ii.  i£ 

10' 

12' 

2210.9  'Sr 

4125.2 

411.  8  |:: 

12' 

14' 

2212.9^ 

4128.3 

412.500 

14' 

16' 

2214.8  «2* 

JH  ^ 

iff 

18' 

2216.7  °"5  . 

4134.5 

4i3.8^£t 

1  8' 

20' 

2218.  6^'S- 

4I37-7 

414.5^- 

20' 

22' 

2220.5  j:  £ 

4140  .8 

415.25=  s 

22' 

24' 

2222.5  <  5 

4I43-9 

4I5-9 

24' 

26' 

2224.4     •£ 

4147.0 

416.6 

26' 

28' 

2226  .3     w 

4150.1 

4I7-3 

28' 

30' 

2228.2     •£ 

4I53-2 

418.0 

3°' 

32' 

2230.2 

4156  .3 

418.7 

32' 

34' 

2232  .  i        R 

4I59-4 

419.4 

34' 

36' 

2234.0       ;§ 

4162.5 

420  .  i 

36' 

38' 

2235.9       < 

4165  .6 

420  .8 

38' 

40' 

2237.8 

4168.7 

421.5 

40' 

42' 

2239.8 

4171.8 

422  .  2 

42' 

44' 

2241.7 

4174.9 

422.9 

44' 

46' 

2243.6 

4178.0 

423.6 

46' 

48' 

2245-5 

4181  .  i 

424.3 

48' 

50/ 

2247.4 

4184.3 

425.0 

5°' 

52' 

2249.4 

4187  .4 

425.7 

52' 

54  - 

2251.3 

4190.5 

426.4 

54' 

56' 

2253.2 

4193.6 

427.1 

56' 

58' 

2255.1 

4196.7 

427.8 

58' 

73 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

43° 

2257.0 

4199.8 

428.5 

43° 

2' 

2259.0 

4202  .9 

429.2 

2' 

4' 

2260  .  9 

4206.0 

429.9 

4; 

6' 

2262  .8 

4209.  I 

430.6 

8' 

2264  .  7 

4212  .  2 

431-3 

8' 

to' 

2266  .6 

4215.3 

432.0 

10' 

12' 

2268.6 

42l8  .4 

432.7 

12' 

14' 

2270.5 

4221  .5 

433-5 

14' 

16' 

2272  .4 

4224.6 

434-2 

16' 

18' 

2274.3 

4227.7 

434-9 

18' 

**' 

2276  .  2 

4230.8 

435-6 

20' 

22' 

2278  .  2 

4133-9 

436.3 

22' 

24' 

2280.1 

4237.0 

437-o 

34' 

26' 

2282  .0 

4240  .  i 

437-8 

26' 

28' 

2283.9 

4243.2 

438.5 

28' 

30' 

2285.8 

4246  .  2 

439-2 

30' 

32' 

2287.7 

4249-3 

439-9 

32' 

34' 

2289.6  ^  ;  4 

4252.4 

440.7    .  .  . 

34' 

36' 

2291   -5     ""'^H 

4255  •  5 

441.4  "><*;: 

36' 

38' 

2293.  4|-     . 

4258.6 

442.1  ez  . 

38' 

40' 

2295.5     rt 

4261  .7 

442.  8  ~ 

40' 

42' 

2297  .6  'g,-  : 

4264  .8 

443  •  5  •£=  : 

42' 

44' 

2299-5^ 

4267.9 

444  .  2  CG 

44' 

46' 

2301.  4  £- 

4271  .0 

445  -0^:  = 

46' 

48' 

2303-3  £4$ 

4274.1 

445-7  £•£;? 

48' 

5°' 

2305.2  ^  **" 

4277.2 

446.4^     M 

5o; 

52/ 

2307.2  j:  : 

4280.3 

447.1  j:  : 

54' 

2309.1  ? 

4283.4 

447-8^ 

54' 

56' 

2311.0 

4286.5 

448.6 

56' 

58' 

2312.9 

4289  .6 

449-3 

58' 

44° 

2314-9 

4292.7 

450.0 

44° 

2' 

2316.8 

4295.8 

450-7 

2' 

4' 

2318.8 

4298.9 

4' 

6' 

2320.7 

4302.0 

452.2 

6' 

8' 

2322.7 

4305  -1 

452.9 

8' 

10' 

2324.6 

4308.1 

453-6 

10' 

12' 

2326  .6 

4311.2 

454-3 

12' 

14' 

2328.5 

4314.3 

455-1 

14' 

16' 

2330-5 

43*7-4 

455-8 

16' 

18' 

2332.4 

4320.5 

456.5 

18' 

20' 

2334-3 

4323-5 

457-3 

20' 

22X 

2336.3 

4326.6 

458.0 

22X 

24' 

2338.2 

4329.7 

458.8 

24' 

26' 

2340.2 

4332.8 

459-5 

26' 

28' 

2342.1 

4335  -Q 

460.3 

28' 

74 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

44°  30' 

2344.0 

4338.9 

461  .O 

44°  30' 

32' 

2346.0 

4342  .0 

461.8 

32' 

34' 

2347v9 

4345  -1 

462.5 

34' 

36' 

2349-8 

4348.2 

463.3 

36' 

38' 

2351-8 

435J-3 

464  .0 

38' 

40' 

2353-7 

4354-4 

464.7 

40' 

42' 

2355-7 

4357-5 

465.5 

42' 

44' 

2357-6 

4360.6 

466.2 

44' 

46' 

2359-6 

4363-7 

467.0 

46' 

48' 

2361.5 

4366.7 

467.7 

48' 

50' 

2363-5 

4369.8 

468.4 

50' 

52' 

2365-5 

4372-9 

469.2 

52' 

54' 

2367-4 

4376.0 

469.9 

-    54' 

56' 

2369.4 

4379-0 

470.7 

56' 

58' 

237I-3 

4382.1 

471.4 

58' 

45° 

2373-3 

4385-2 

472.1 

45° 

2' 

2375-3 

4388.3 

'    472.8 

2' 

4' 

2377-2  t&  + 

4391-4 

473-6    .  .  . 

4' 

6' 

2379.2.-.     M 

4394.5 

474-3  ^     H 

6' 

8' 

2381.2^:  : 

4397-6 

475-1  j°s  : 

8' 

10' 

2383.il 

4400  .6 

475-8-g 

10' 

12' 

2385-  *  £"  : 

4403.7 

476.6-g=: 

12' 

14' 

2387.0^ 

4406  .  8 

477-3^ 

14' 

16' 

2389.0^- 

4409.9 

478.  i£-  : 

16' 

18' 

o  oo  oo 

2390'9       M     NXA 

ui  v,  O 

4413.0 

478.8^ 

18' 

20' 

2392.9^ 

4416  .0 

479-6-d 

20' 

22' 

2394.95:   : 

4419.0 

480.  43:  : 

22' 

24' 

2396.  8^ 

4422.1 

481.1 

24' 

26' 

2398.8 

4425.2 

481.9 

26' 

28' 

2400  .  7 

4428.3 

482.7 

28' 

30' 

2402.7 

4431-3 

483-4 

30; 

32' 

2404.7 

4434-4 

484.2 

32 

34' 

2406  .6 

4437-5 

485.0 

3i 

36' 

2408.6 

4440  .  5 

485-7 

36 

38' 

2410.5 

4443-6 

486.5 

38' 

40' 

2412.5 

4446  .  7 

487.2 

40' 

42' 

2414.5 

4449  -  8 

488.0 

42' 

44' 

2416  .4 

4452.8 

488.8 

44' 

46' 

2418.4 

4455-9 

489.5 

46' 

48' 

2420.3 

4459-0 

490.3 

48' 

50' 

2422.3 

4462  .  i 

491  .0 

50' 

52: 

2424-3 

4465.1 

491.8 

52 

54'. 

2426  .  2 

4468.2 

492.5 

54 

56' 

2428.2 

4471-3 

493-3 

56 

;  58' 

2430.1                 4474-4 

494.1 

58' 

75 


IX.-— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

46° 

2432.1 

4477-5 

494-8 

4G° 

2' 

2434.1 

4480  .  6 

495-6 

2' 

4' 

2436.0 

4483.6 

496.3 

4' 

6' 

2438.0 

4486.7 

497.1 

6' 

r 

2439.9 

4489.8 

497-9 

8' 

10' 

2441.9 

4492.8 

498.7 

10' 

12' 

2443.9 

4495-9 

499-5 

12' 

14' 

2445.8 

4498.9 

5°o-3 

14' 

16' 

2447.8 

4502  .0 

501  .0 

16' 

18' 

2449.7 

4505.1 

501.8 

18' 

20' 

2451-7 

4508.  i 

502.6 

20' 

22' 

2453-7 

4511.2 

503.4 

22' 

24' 

2455-7 

45*4.3 

504-2 

24' 

26' 

2457.6 

45*7-3 

504.9 

26' 

28' 

2459.6 

4520.4 

505.7 

28' 

30' 

2461  .6 

4523-4 

506.5 

•      3°' 

32' 

2463.6 

507-3 

32' 

34' 

2465-5  »A  o,4 

4529.6 

508.0 

34' 

36' 

2467.5 

4532.6 

508.8  ^** 

36' 

38' 

2469-  5  £:  : 

4535-7 

509-6  6,  : 

38' 

.40' 

247I-5l  . 

4538.8 

510.4* 

40' 

42' 

2473.51"  " 

4541.8 

5II.  2.  |:  = 

42' 

44' 

2475-4  g.  - 

4544-9 

512  .0  co 

44' 

46' 

2477.  4  ^  a"H 

4548.0 

512  .7  ,0=    5 

46' 

48' 

2479-  4  ~«o 

455i.o 

5*3-5  ^^ 

48' 

50' 

2481.4.?*" 

4554.1 

5*4-3  s*t£ 

50' 

52 

2483.45=  = 

4557-2 

515  .  i  •tfs  = 

52' 

54' 

2485.4 

4560.2 

515  .9 

54' 

56' 

2487-3 

4563.3 

516.6 

56' 

58' 

2489.3 

4566.4 

5*7-4 

58' 

47° 

2491.3 

4569-4  , 

518.2 

47° 

2' 

2493.3 

4572.4 

519.0 

2' 

4' 

2495.3 

4575-5 

519.8 

4' 

6' 

2497.3 

4578.5 

520.6 

6' 

8' 

2499.2 

4581.6 

521.4 

8' 

10' 

2501  .2 

4584.6 

522  .  2 

10' 

12' 

2503.2 

4587-7 

523  "O 

12' 

14' 

2505.2 

4590.7 

523.8 

14' 

16' 

2507.2 

4593-8 

524.6 

16' 

18' 

2509.1 

4596.8 

525.4 

18' 

20' 

2511  .  I 

4599-9 

526  .  2 

20' 

22' 

25I3.I 

4602  .  9 

527.0 

22' 

24' 

25*5.  i 

4606  .0 

527.8 

24' 

26' 

2517.1 

4609  .  o 

528.6 

26' 

28' 

2519.1 

4612  .  i 

529-4 

28' 

76 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

47°  3°' 

2521  .O 

4615.1 

530.2 

4:7°    30' 

32' 

2523.0 

4618.2 

53LO 

32' 

34' 

2525-0 

4621  .2 

53L8 

34' 

36' 

2527  .0 

4624.3 

532.6 

36' 

38' 

2529.0 

4627.3 

533-4 

38' 

4°' 

2531.0 

4630.4 

534-2 

40' 

42' 

2533.0 

4633.4 

535-0 

42' 

44' 

2535.0 

4636.5 

535-8 

44' 

46' 

2537.0 

4639.5 

536.6 

46' 

48' 

2539.0 

4642  .6 

537-4 

48' 

5°' 

2541.0       4 

4645-6 

538.2 

50/ 

52' 

2543.0 

4648.7 

539-0 

52! 

54' 

2545.0      £ 

4651.7 

539-8 

54 

56' 

2547-0        -3 

4654-8 

540.6 

56' 

58' 

2549.0        .£ 

4657.8 

54L4 

58' 

48° 

CO 

2551-0        K 

4660  .9 

542.3 

48° 

2' 

2553-0       * 

4663.9 

543-1 

2' 

4' 

2555-0       o 

4667  .0 

544-0    .  .  . 

4' 

6' 

2557.0  ^<*£ 

4670.0 

544.8  ""*;? 

6' 

8' 

2559-0  |s  :§ 

4673-1 

545-    6         0;        = 

8' 

10' 

2561.0*2 

4676  .  i 

546.  4-g 

10' 

12' 

2563  -o-a: 

4679.1 

547.  2  -a:  : 

12' 

14' 

2565-0^ 

4682.2 

548.0^ 

14' 

1  6' 

2567.0^ 

4685.2 

548.9-5=  = 

16' 

18' 

4688.2 

549-7^25 

18' 

20' 

2571-0  jpd 

4691.3 

55°-5-;g_  _H 

20' 

22' 

2573-o^:  * 

4694.3 

22' 

24' 

2575-0^    "g 

4697.3 

552.2 

24' 

26' 

2577-0        £ 

4700.4 

553-o 

26' 

28' 

2579.0         g 

4703.4 

553-8 

28' 

3°' 

2581.0        *£> 

4706.4 

554-6 

30; 

32' 

2583-0        ^ 

4709.5 

555-4 

32 

34' 

2585-0        £ 

47I2-5 

556.2 

34 

36' 

2587.0      3 

47^.5 

557-0 

36' 

38' 

2589.0 

4718.6 

557-9 

38' 

40' 

2591  .0 

4721.6 

558.7 

40' 

42' 

2593-0 

4724.6 

559-5 

42' 

44' 

2595-0 

4727.6 

560.3 

44' 

46' 

2597-1 

4730.7 

561.2 

46' 

48' 

2599.1 

4733-7 

562.0 

48' 

50' 

2601  . 

4736.8 

562.8 

5°' 

52' 

2603. 

4739-8 

563-6 

52', 

54' 

2605. 

4742.8 

564-4 

54 

56' 

2607  . 

4745-9 

565-3 

56 

58' 

2609. 

4748.9 

566.1 

58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

49° 

26ll  .2 

4752.0 

566.9 

49° 

2' 

2613  .2 

4755-0 

567-7 

2' 

4' 

2615.2 

4758'.© 

568.6 

4' 

6' 

2617.2 

4761  .0 

569-4 

6' 

8' 

2619  .  2 

4764.  i 

57°-3 

8' 

10' 

2621  .  2 

4767.1 

571-1 

10' 

12' 

2623.3 

4770.1 

572.0 

12' 

14' 

2625.3 

4773-1 

572.8 

14' 

16' 

2627.3 

4776.2 

573-6 

16' 

1  8' 

2629.3 

4779-2 

574-5 

i8r 

20' 

2631.3         4 

4782.3 

575-3 

20' 

22' 

2633.4 

4785-3 

576.2 

22' 

24' 

2635.4        | 

4788.3 

577-o 

24' 

26' 

2637.4        - 

4791.4 

577-9 

26' 

28' 

2639.4        .£ 

4794.4 

558.7 

28' 

30' 

2641.4        ^ 

4797-4 

579-5 

30' 

32' 

2643.5         £ 

4800.5 

580.4 

32' 

34' 

2645-5     5 

4803.5 

581.2    .  .  . 

34' 

36' 

2647.5  xAoR 

4806  .  5 

582.1  IOCV2 

36' 

38' 

2649.5  |:  ? 

4809  .6 

582.9^,      ; 

& 

40' 

2651.5-3     ^ 

4812.6 

583-8^ 

40' 

42' 

2653.6-*: 

4815.6 

584.  6.  g:  : 

42' 

44' 

2655.6* 

4818.6 

585.5^ 

44' 

46' 

2657  .6  ,0  = 

4821.7 

sWvs'l;  = 

46' 

48' 

2659.6  <^<^^ 

4824.7 

587  .2  oq  e>» 

48' 

So' 

2661  .6  £'•£  M 

4827.7 

588.!^" 

50' 

$*' 

2663.  7  5.  ;§ 

4830.7 

588.95=   = 

52' 

54' 

2665.7  ^    *§ 

4833-8 

589-  7< 

54' 

56' 

2667.7      -g, 

4836.8 

590.6 

56' 

58' 

2669.7      <« 

4839.8 

59i-4 

58' 

5O° 

2671.7      <~ 

4842.9 

592.3 

5O° 

2' 

2673.7      ^§ 

4845-9 

593-2 

2' 

4' 

2675.8      •*• 

4848.9 

594-0 

4r 

6' 

2677.8      S 

4852.0 

594-9 

6' 

r 

2679.8 

4855-0 

595-8 

8' 

10' 

2681.9 

4858.0 

596.7 

10' 

12' 

2683.9 

4861  .0 

597-6 

12' 

14' 

2686.0 

4864.0 

598.4 

14' 

16' 

2688.0 

4867. 

599-3 

16' 

18' 

26^0.  i 

4870. 

600.  i 

18' 

20' 

2692  .  i 

4873- 

601  .0 

20' 

22' 

2694.  i 

4876. 

601  .9 

22' 

24' 

2696  .  2 

4879. 

602  .7 

24' 

26' 

2698.2 

4882  .2 

603.6 

26' 

28' 

27OO  .3 

4885.2 

604.4 

28' 

78 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 

Chord. 

External 
Secant. 

Whole 
Angle. 

50°  30' 

2702.3 

4888.2 

605-3 

50°  30' 

32' 

2704.3 

4891  .  2 

606  .  2 

32' 

34' 

2706  .4 

4894  .  2 

607  .O 

34' 

36' 

2708  .4 

4897.2 

607.9 

36' 

38' 

2710.5 

49OO  .  2 

608.8 

38' 

4°' 

2712.5 

4903.2 

609.7 

40' 

42' 

27M.5           - 

4906  .  2 

610  .  6 

42' 

44' 

2716.6          c 

4909.2 

611.5 

44' 

46' 

2718.6         £ 

4912.2 

612.3 

46' 

48' 

2720.7          | 

p, 

49I5-2 

613-2 

48' 

5°' 

2722  .  7        ^ 

4918  .  2 

614.  i 

50' 

52' 

2724.8       £ 

4921.3 

615  .0 

52' 

S4' 

2726.8        <? 

4924.3 

615.8 

54' 

56' 

2728.8        | 

4927-3 

616.7 

56' 

58' 

2730.9       1" 

4930-3 

617.6 

58' 

51° 

2732.9       < 

4933.3 

618.5 

51° 

2' 

2734-9 

4936.3 

619.4 

2' 

4' 

2737.0 

4939-3 

620.3    •  •  . 

4' 

6' 

2739.0 

4942.3 

621.2  l"°:r 

6' 

8' 

2741  .  1   ">  c>4 

4945-3 

622  .0  c:  - 

8' 

10' 

2743.1  |  =  : 

4948.3 

622.  Q-g 

10' 

12' 

2745.1- 

495J-3 

623.8-^= 

12' 

14' 

2747.  2.  £:  - 

4954-3 

624.  7<£ 

14' 

16' 

2749.2  & 

4957-3 

625.6,5=  = 

16' 

1  8' 

275J-3.C:  : 

4960.3 

626.4  °^  " 

^     W  00  00 

18' 

20' 

2753-3  2S2. 

4963-3 

627-  3-d 

20' 

22' 

2755-3  £3R 

4966.3 

628.233-  = 

22' 

24' 

2757-4;^  s 

4969.3 

629  .  1 

24' 

26' 

2759-4<T  " 

4972-3 

630.0 

26' 

28' 

2761.5 

4975-3 

630.8 

28' 

3°' 

2763.6 

4978.3 

631-7 

3o' 

32' 

2765.6         j 

4981.3 

632.6 

32' 

34' 

2767.7         o 

4984-  3 

633.5 

34' 

36' 

2769.8        * 

4987-3 

634.4 

36' 

38' 

2771.8        *£ 

4990-3 

635-3 

38' 

40' 

2773.9         c& 

4993-3 

636.2 

40' 

42' 

2775.9      £ 

4996.3 

637-1 

42' 

44' 

2778.0 

4999-3 

638.0 

44' 

46' 

2780.  I 

5002.3 

638.9 

46' 

48' 

2782.2       £ 

5005.3 

639.8 

48' 

5°' 

2784.2       < 

5008.3 

640.7 

50' 

52 

2786.3 

50H-3 

641  .6 

52' 

54' 

2788.4 

5014-3 

642.5 

54' 

56' 

2790.4 

5OI7-3 

643-4 

56' 

58' 

2792-5 

5020  .3 

644-3 

58' 

79 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

52° 

2794.5 

5023.3 

645.2 

>2° 

2' 

2796.6 

5026.3 

646.  I 

2' 

4' 

2798.6 

5029.3 

647.0 

4' 

6' 

2800.7          4 

5032  .  3 

647.9 

6' 

8' 

2802.7 

o 

5035-3 

648  .  8 

8' 

10' 

2804.8        5z 

5038.3 

649-7 

10' 

12' 

2806.8        1J» 

504L3 

650.6 

12' 

14' 

2808.9        '& 

5044-3 

651  .  5 

14' 

16' 

28ll    .O                   VH 

5047.3 

652.4 

16' 

18' 

28I3.I                 <~ 

5050-3 

653-3 

18' 

20' 

2815.2  ^  ^  £ 

5053.3 

654.3 

20' 

22' 

2817  .3  ?o  4J 

5056.3 

655  •  2 

22' 

24' 

2819.  3  1'  ^ 

5059-3 

656.1 

24' 

26' 

2821  .4  'g 

5062.3 

657.0 

26' 

28' 

2823.5  ja: 

5065.3 

658.0 

28' 

3°' 

2825.5  &. 

5068.3 

658.9 

3°' 

32' 

2827  .6  ^  -r 

507L3 

659.8 

32' 

34' 

2829.7  H.O  4 

5074.3 

660.7      .    .    . 

34' 

36' 

2831.7,5*0 

5077.3 

66  1  .6  "'°*'* 

36' 

38' 

2833.8^:  fc 

5080.3 

662.5  o.  : 

38 

40' 

2835.9  '  -| 

5083.3 

663.5-3 

40' 

42' 

2837.9         co 

5086.3 

664.  4  £5  = 

42' 

44' 

2840.0       J5 

5089-3 

665.3^ 

44' 

46' 

2842.1        « 

5092.3 

666  .3  £-  : 

46' 

48' 

2844.2           S" 

5095-3 

667-2  ^^^ 

48' 

50' 

2846.3      ;§ 

5098.3 

668.1^     H 

50' 

52' 

2848.4        <! 

5IOI-3 

669.0^=  s 

52' 

54' 

2850.4 

5I04-3 

670  .0 

54' 

56' 

2852.5 

5I07-3 

670.9 

56' 

58' 

2854.6 

5110.2 

671.8 

58' 

53° 

2856.7 

5113.2 

672.7 

53° 

2' 

2858.8  t^4 

5116.2 

673-6 

2' 

4' 

2860.8    .      M 

5119.2 

674.6 

4' 

6' 

2862.9  £:  : 

5122  .  2 

675-5 

6' 

8' 

2865.0-3 

5I25-i 

676.4 

8' 

10' 

2867.  1  'J.5  : 

5128.1 

677-3 

10' 

12' 

2869.2        *               tH 

5131  .  i 

678.2 

12' 

14' 

287  1-  3^V^> 

5134.  i 

679.1 

14' 

16' 

2873.4  ^£S- 

5I37-° 

680.  i 

16' 

18' 

2875.5^^ 

5140  .0 

681.0 

18' 

20' 

2877.S3-  : 

5142.9 

682.0 

20' 

22' 

2879.6^ 

5J45-9 

682.9 

22' 

24' 

2881.7 

5148.9 

683.8 

24' 

26' 

2883.8 

5151-8 

684.8 

26' 

28' 

2885.8 

5J54-8 

685.7 

28' 

80 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

53°  30' 

2887.9 

5157.8 

686.7 

53°  30' 

32' 

2890.0 

5160.8 

687.6 

32' 

34' 

2892  .  I 

5163.7 

688.6 

34' 

36' 

2894.2  xA^4 

5166.7 

689.5 

36' 

38' 

2896.3  6 

5l69-7 

690.5 

38' 

40' 

2898.  4*' 

5172.6 

691.4 

40' 

42' 

2900.5  *   : 

5175  .6 

692.4 

42' 

44' 

2902  .6  jg" 

5178.6 

693.3 

44' 

46' 

2904-7  5-  : 

694.3 

46' 

48' 

2906.8  ^^ 

5184.5 

695-2 

48' 

50' 

2908.9  £££ 

5187.4 

696.  i 

50' 

S^' 

2911.0^* 

5190.4 

697.1 

52' 

54' 

2913.1  3=  s 

5I93.4 

698.0 

54 

56' 

2915.2 

5196.3 

699  .0 

56 

58' 

5*99-3 

700  .0 

58' 

54° 

2919.4 

5202.3 

700.9 

54° 

2' 

2921.5 

5205.3 

701.9 

2' 

4' 

2923.6 

52O8  .  2 

702.8 

4' 

6' 

2925.7 

5211  .  2 

703.8  ^j 

6' 

8' 

2927  .8 

5214.2 

704.7       0;      - 

8' 

10' 

2929.9 

52I7.I 

705.713 

10' 

12' 

2932.0 

5220.1 

706.6.*:  : 

12' 

14' 

2934.1 

5223.1 

707  .6  co 

I4f 

1  6' 

2936.2 

5226  .O 

708.5,°=  = 

16' 

1  8' 

2938.3       .  . 

5229.0 

709.5  J££ 

18' 

20' 

2940.4  ^- 

5232.0 

710.5  ^  ~  M 

20' 

22' 

2942.51:  : 

5235.0 

711.  4|=  : 

22' 

24 

2944.  6  ~ 

5237.9 

712.4 

24' 

26' 

2946.7.*:  : 

5240.9 

7I3.4 

26' 

28' 

2948.8  co 

5243.9 

7J4.3 

28' 

3°' 

2950.9^  : 

5246.8 

715.3 

3°' 

32' 

2953.0^2 

5249.8 

716  .3 

32; 

34' 

2955.2  £3° 

5252.8 

717.2 

34 

36' 

2957.  3I§-  s 

718.2 

3o 

38' 

2959.  4<"  " 

5258^ 

719.2 

38' 

40' 

2961.5 

5261  .6 

720  .  i 

40' 

42' 

2963.6 

5264.6 

721.1 

42' 

44' 

2965.8 

5267.6 

722  .1 

44' 

46' 

2967.9 

5S70.5 

723.1 

46' 

48' 

2970.0 

5273.5 

724.0 

48' 

5°' 

2972.1 

5276.5 

725.0 

50/ 

52' 

2974.2 

5279.5 

726  .0 

$2', 

54' 

2976.4 

5282.4 

727.0 

56' 

2978-5 

5285.4 

728.0 

56' 

58' 

2980  .  6 

5288.3 

728.9 

58' 

81 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant  . 

Whole 
Angle. 

55° 

2982.7 

529I-3 

729.9 

55° 

2' 

2984.8 

5294.2 

73°-9 

2' 

4' 

2987  .0 

5297.2 

731-9 

4' 

6' 

2989.1          4 

5300.1 

732.  -9 

6' 

8' 

2991.2          £ 

5303-I 

733-8 

8' 

10' 

2993.3         * 

5306.0 

734-8 

10' 

12' 

2995.4        .§ 

5309-0 

735-8 

I2f 

14' 

2997.6         £ 

5311  -9 

736.8 

14' 

16' 

2999-7          £ 

53J4-9 

737-7 

16' 

18' 

3001.8         £ 

53I7-8 

738.7 

18' 

20' 

3003.9  ^o 

5320.8 

739-7 

20'. 

22' 

3006.0   6_   £ 

740.7 

22r 

24' 

3008.2  K~  J5 

5326.7 

74L7 

24' 

26' 

3°IO-3  I? 

5329V.6 

26' 

28' 

3012.4  £: 

5332.6 

743-6 

28' 

30; 

3014.5   S: 

5335-5 

744-6 

30' 

3016.6     ^ 

5338.5 

745-6 

32' 

34J 

3018.8  ~£? 

534L4 

746.6  ^4 

34' 

36' 

3020.9^*6 

5344.4 

747-6    . 

36' 

38' 

3023.  o|:  J 

5347-3 

748.6^: 

38' 

40' 

3025.1          1 

5350.2 

749.6^  j 

40' 

42' 

3027.3          to 

5353-2 

750.  6  |r  ' 

42' 

44' 

3029.4         .0 

5356.1 

751-6  £.  5 

44' 

46' 

3031.5         «r 

5359-1 

752  .6  ^w  ^ 

46' 

48' 

3033.7         o 

5362.0 

753.6    ci^oo 

48' 

5o; 

3035-8        :§ 

5365-0 

754-6^:  = 

50/ 

3037-9        < 

5367.9 

52' 

54' 

3040.0 

5370.9 

75^6 

54 

56' 

3042.2 

5373-8 

757-6 

56' 

58' 

3044-3 

5376.8 

758.6 

58' 

56° 

3046.5 

5379-7 

759-6 

56° 

2f 

3048.6  ^£4 

5382.7 

760.6 

2' 

4' 

3050.8    .     M 

5385-6 

761.6 

4' 

6' 

3052.9^"  K 

5388.6 

762.6 

6' 

8' 

3055.1  -j^  _ 

5391-5 

763.6 

8' 

10' 

3057-2'^"  : 

5394-4 

764.6 

ior 

12' 

3059-2  g.  .- 

5397-4 

765:6 

12' 

14' 

3061.  4^00 

5400.3 

766.7 

14' 

16' 

3063.6       MC^g. 

5403-2 

767.7 

16' 

18' 

3065.8       «     •*«- 

54o6.2 

768.7 

18' 

20' 

3067.9^-  = 

5409.1 

769-7 

20' 

22' 

3070.0 

5412.0 

770.7 

22' 

24' 

3072.2 

5415  .0 

771.7 

24' 

26' 

3074.4 

54I7-9 

772.8 

26' 

28' 

3076.5 

5420  .  8 

773-8 

28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

56°  30' 

3078.6 

5423.7 

774-8 

56°  30' 

32' 

3080.8 

5426.7 

775-8 

32' 

34' 

3082.9 

5429.6 

776.8 

34' 

36' 

3085.1  t*4 

5432.5 

777-9 

36' 

38' 

3087.3  ^  ^ 

5435-5 

778.9 

38' 

40' 

3089.  4  2" 

5438.4 

779-9 

40' 

42' 

3091  .6  g.  - 

544L3 

780.9 

42' 

44' 

3093.  8  '|T  • 

5444.3 

782.0 

44' 

46' 

3095.9  fc:  - 

5447-2 

783-0 

46' 

48' 

309S.o-^ 

5450.1 

784.0 

48' 

50' 

3100.1  i%$ 

5453-1 

785.0 

5°' 

52' 

3102-  3-c 

5456.0 

786.0 

52' 

54' 

3104.  4^:  = 

5458.9 

787.0 

54' 

56' 

3106  .6 

5461.9 

788.1 

56' 

58' 

3108.7 

5464.8 

789.1 

58' 

57° 

3110.9 

5467.8 

790.1 

57° 

2' 

5470.7 

791.2 

2' 

4' 

3115-2 

5473-6 

792.2 

4' 

6' 

3II7-4 

5476.6 

793.2   ">*•+ 

6' 

8' 

3119.6         6 

5479-5 

794.3   d-  : 

8' 

io' 

3121.7         -2 

5482.4 

795-3^ 

10' 

12' 

3123.9        •£ 

5485.3 

796.3.*:  : 

12' 

14' 

3126.0     «« 

5488.3 

797-4^ 

14' 

16' 

3128.2         ,0 

5491.2 

798.4J3--  : 

16' 

18' 

3130.4          ? 

5494-1 

799-5  <*«.^. 

18' 

20' 

3132.5  >?^£ 

5497.0 

wocco 
800.5 

20' 

22' 

5499-9 

80  1  .  5  •d:  : 

22' 

24' 

3i36.'82~3 

5502.8 

802.  6  < 

24' 

26' 

5505-8 

803.6 

26' 

28' 

3141  .2  C& 

5508.7 

804.7 

28' 

30' 

3143.3!' 

5511  .6 

805.7 

30; 

32' 

3145.5       M     CO* 

55*4.6 

806.7 

32 

34' 

3147.6  S3" 

5517  .  5 

807.8 

i      34 

36' 

3149-85,  £ 

5520.4 

808.8 

36 

38' 

3152.  o<f  -g 

5523.4 

809  .9 

:      38' 

40' 

3154.2         £ 

5526.3 

810.9 

40' 

42' 

3156.4        £ 

5529.2 

811  .9 

42' 

44' 

5532.1 

813.0 

44' 

46' 

3160.7        <5 

5535.o 

814.0 

46' 

48' 

3162.9 

5538.0 

815.1 

48' 

50' 

3165.1        | 

5540.9 

816.1 

50/ 

52' 

3167.2 

5543-8 

817.2 

52/ 

54' 

3169.4 

5546.7 

818.2 

56' 

3171.6 

5549-7 

819.3 

56' 

58' 

3I73-8 

5552.6 

820.3 

58' 

83 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

58° 

3176.0 

5555-5 

821.4 

58° 

2' 

3178.2 

5558.4 

822.4 

2' 

4' 

3180.4 

5561.3 

823.5 

4' 

6' 

3182.6          4 

5564.2 

824.6    . 

6' 

8' 

3184.8          £ 

5567.1 

825.6  ^S 

8' 

10' 

3186.9        55. 

5570.0 

826.  7»g: 

ior 

12' 

3189.1         *g 

5573-0 

827.713 

12' 

14' 

3I9L3         $• 

5575-9 

828.8-^: 

14' 

16' 

3193-5      -3 

5578.8 

829.9^ 

16' 

ft' 

3r95-7        ^ 

831.0!= 

18' 

20' 

3197.8    .  .<§ 

5584.6 

832.0^5 

20'. 

22' 

3200.0  j?0^ 

5587  .  5 

S33.i^M 

22' 

24' 

3202.2  ^:  5 

5590.4 

834-15= 

24' 

26' 

3204.  4g 

5593-3 

835-2^ 

26' 

28' 

3206  .6  'p.: 

5596.2 

836.2 

28' 

32' 

3208.7   £. 

5599-1 

837.3 

30' 

3  2  1  o  .  9  ^  c, 

5602  .0 

838.3 

32' 

:   34; 

3213-1  d£? 

5604.9 

839.4 

34' 

36 

32i5-3^6 

5607.8 

840.5 

36' 

*  38' 

3217-5^=  * 

5610.7 

841.6 

38' 

40' 

3219.7    4 

5613-6 

842.7 

40' 

42' 

3221.9        m 

5616.6 

843-8 

42' 

44' 

3224.1        .0 

5619  .  5 

844.9 

44' 

46' 

3226.3         «? 

5622.4 

845-9 

46' 

48' 

322*.  5       1 

5625.3 

847.0 

48' 

50/ 

3230.7     3 

5628.2 

848.1  ^i 

50' 

52' 

3232.9      <: 

5631.  i 

849-2  6.  . 

52; 

54' 

3235.! 

5634-0 

850.  3g~ 

56' 

3237-3 

5636-9 

851  .4  *§ 

56' 

58' 

3239.5 

5639-8 

852.4  £:= 

58' 

59° 

3241-7 

5642.7 

853.5  !-  : 

59° 

2' 

3243.9  ^4 

5645.6 

854  '  6     O\  fOOO 

2' 

4' 

3246.1      H 

5648.5 

855.6    «••«« 

4' 

6' 

3248.  3£= 

565L4 

856.73,    - 

6' 

8' 

3250.5^ 

5654.3 

857-  8<"  " 

8' 

10' 

3252.  7  '^ 

5657.2 

858.9 

10' 

12' 

3254.9  g. 

5660  .  i 

860.0 

12' 

14' 

3257-1^ 

5663.0 

861.1 

14' 

16' 

3259.3  ^ 

5665.9 

862.2 

16' 

18' 

3261.5  *«- 

5668.8 

863.3 

18' 

20' 

3263.75= 

5671-7 

864.4 

20' 

22' 

3266.0 

5674.6 

865.5 

22' 

24' 

3268.2 

5677.5 

866.6 

24' 

26' 

3270.4 

5680.4 

867.7 

26' 

28' 

3272.6 

5683.3 

868.8 

28' 

84 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 

Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

59°  30' 

3274.8 

5686.2 

869.9 

59°  30' 

32' 

3277.0 

5689.1 

871.0 

32' 

34' 

3279.2 

5691.9 

872.1 

34' 

36' 

3281.4          2 

5694.8 

873.2 

36' 

38' 

3283-6          6 

5697.7 

874.3 

38' 

40' 

3285-8         i 

5700.6 

875.4 

40' 

42' 

3288.0        •% 

5703.5 

42' 

44' 

3290.2        w 

5706.4 

877*6 

44' 

46' 

3292.4        £ 

5709.3 

878.7 

46' 

48' 

3294.7         ^ 

5712.2 

879.8 

48' 

5°' 

3296.9 

57I5-1 

880.9          j 

50' 

52' 

3299.1         3 

57J7-9 

882.0          6 

52' 

54' 

330L4 

5720.8 

883.1         2 

54' 

56' 

3303.6 

5723-7 

884.2          g 

56' 

58' 

3305.8 

5726.6 

885.3      £ 

58' 

6O° 

33o8.o 

5729-6 

886.4       & 

6O° 

2' 

33IO-3 

5732.5 

887.5           oo 

2' 

4' 

3312.5  -.o4 

5735.4 

888.6    .  .^ 

4' 

6' 

3314.7  6-  - 

5738.2 

889.7  "/^ 

6' 

8' 

3316.  9|zr  - 

5741  -1 

890.9^=  < 

8' 

10' 

3319-  I  |:  : 

5744.0 

892.0^, 

10' 

12' 

3321.  3c£ 

5746.9 

893-1^ 

12' 

14' 

3323-5  c:  : 

5749.8 

894.2  g. 

14' 

1  6' 

3325-7  ">qvo 

5752.7 

895.4^^ 

16' 

18' 

3328.0  ZZ^ 

5755-5 

896.  5  ^06 

18' 

20' 

3330.2^^ 

5758.4 

897.65=  r 

20' 

22' 

3332.  4«f  ' 

5761  .3 

898.7    '      6 

22 

24' 

3334-7 

5764.2 

899.8         * 

24' 

26' 

3336.9 

5767.1 

901.0          § 

26' 

28' 

3339-1 

5769-9 

902.1         J| 

28' 

30' 

3341.4 

5772.8 

903.2        J5 

30' 

32' 

3343-6         ? 

5775-7 

904.3           o, 

32 

34' 

3345-9         6 

5778.6 

905.4    s 

34' 

36' 

3348.1        * 

906.6    ^ 

36' 

38' 

3350-3        | 

5784.3 

907.7    ^ 

38' 

40' 

3352-6      £ 

5787-2 

908.8 

40' 

42' 

3354-8         £ 

579o.i 

910.0 

42' 

44' 

3357  -1         - 

5793-0 

911.1 

44' 

46' 

3359-3        *§ 

5795-9 

912.2 

46' 

48' 

336i.5        £ 

5798.7 

9J3-3 

48' 

s°; 

3363.8        ? 

5801  .6 

9J4-5 

5°' 

3366.0 

5804.5 

915.6 

52', 

54' 

3368.3 

5807.4 

916.7 

54 

56' 

3370.5 

58lO.  2 

917.9 

56 

58' 

3372-8 

58I3.I 

919.1 

58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

61° 

3375-0 

5816.0 

92O  .  2 

61° 

2' 

3377-3 

5818.9 

921.3 

2' 

4' 

3379-5 

5821.7 

922.5 

4' 

6' 

5824.6 

923.6 

6' 

8' 

3384-0        : 

5827.5 

924.8 

V 

10' 

3386.2        * 

5830.3 

925.9 

10' 

12' 

3388.5         g 

5833.2 

927.0 

12' 

14' 

3390.7        £ 

5836.0 

928.2 

14' 

16' 

3393-0        g 

5838.9 

929.4 

16' 

18' 

3395-2        £ 

5841.8 

930.5 

18' 

20' 

3397-5         o 

5844.7 

931.6     .    . 

20'. 

22' 

3399-8        ^ 

5847  .  6 

932.8      '       - 

22' 

24' 

3402.0        Jj 

5850-4 

933-9;°-  = 

24' 

26' 

3404-3 

5853.3 

935  -1  -3- 

26' 

28' 

3406.5 

5856.2 

936.3-^  : 

28' 

30' 

3408.8 

5859.0 

937-4? 

3o' 

32' 

3411.1 

5861.9 

938.  6£:  : 

32' 

34' 

3413.3  ^.&  + 

5864-8 

939-7  ?5od 

34' 

36' 

34I5.6  6^   " 

5867-6 

940.9  ^ 

36' 

38' 

5870.5 

942.1  ^:  : 

38' 

40' 

3420.1  |,  , 

5873.3 

943-2 

40' 

42' 

3422.  4  £~  " 

5876.2 

944-4 

42' 

44' 

3424.6  £:  : 

5879.1 

945  •  5 

44' 

46' 

3426.  9  ^H  c* 

5882.0 

946.7 

46' 

48' 

3429-1  £*« 

5884.8 

947-9 

48' 

50' 

343.1.  4  -£* 

5887.7 

949-o 

50' 

52' 

3433-  7  <:  : 

5890.6 

950.2 

52' 

54' 

3435-9 

5893  .4 

95J-3 

54' 

56' 

3438.2 

5896.3 

952.5 

56' 

58' 

3440  .  5 

5899.1 

953-6 

58' 

62° 

3442.7 

5902  .0 

954.8 

62° 

2' 

3445-0         4 

5904.8 

956.0    .  .  . 

2' 

4' 

3447-2         £ 

5907-7 

957.2  l°°? 

4' 

6' 

3449-5        £ 

59io.5 

958.3   o":  = 

6' 

8' 

3451-8        -g 

59*3-4 

959.5^ 

8' 

10' 

3454-1        £ 

5916  .  2 

960.7  -^:  : 

10' 

12' 

3456.4        £ 

59i9.i 

961.9^ 

12' 

14' 

3458.6        - 

592L9 

963.1,0:  : 

14' 

16' 

3460.9         o> 

5924.8 

16' 

18' 

3463.2 

5927-6 

965.4^°°^ 

18' 

20' 

3465.5        ^ 

5930.5 

966  .6  J:  : 

20' 

22' 

3467-8 

5933-3 

967.8    ' 

22' 

24' 

3470.0 

5936.1 

969.0 

24' 

26' 

3472.3 

5939-0 

970.1 

26' 

28' 

3474-6 

5041  -8 

071  -3 

28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

02°  30' 

3476.9 

5944-7 

972.5 

62°  30' 

32' 

3479-2 

5947-5 

973-7 

32' 

34' 

348i.5 

5950-4 

974-9 

34' 

36' 

3483.8  ^^  + 

5953-2 

976.0 

36' 

38' 

3486.0  6 

5956.1 

977-2 

38' 

40' 

3488.  3*: 

5958.9 

978.4 

40' 

42' 

3490.6  g.  „ 

596i.8 

979-6 

42' 

44' 

3492.  9  £-  ' 

5964.7 

980.8 

44' 

46' 

3495-2  g.  . 

5967-5 

982  .0 

46' 

48' 

3497  •  5  "£„  H 

5970.4 

983.1          4 

48' 

5o' 

3499.7  £2£ 

5973-2 

984.3        £ 

50' 

S*' 

3502.0^ 

5976.1 

985-5     2 

52' 

54' 

3504-  2^    = 

5978.9 

986.7      a  . 

54' 

56' 

3506.5 

598i.8 

987-9     1 

56' 

58' 

3508.8 

5984.6 

989.1        £ 

58' 

63° 

3511-1 

5987-4 

990.3 

63° 

2' 

3513.4 

5990.3 

991.5                      H 

2' 

4' 

35*5-7 

5993-1 

992.7  .  .3 

4' 

6' 

35*8.  o 

5996.0 

993-9  c-    < 

6' 

8' 

352o.3 

5998.8 

995-  i  £: 

8' 

10' 

3522.6 

6001  .6 

996.  3.|- 

10' 

12' 

3524.9 

6004  .4 

997-  5£ 

12' 

14' 

3527-2 

6007  .  2 

998.7  fcs 

14' 

,  16' 

3529'5 

6OIO  .  I 

999-9^ 

16' 

18' 

3531-8 

6OI2  .9 

IOOI  .  I    ^06 

18' 

20' 

w?  O>  "* 

3534-1     ;      H 

6015.8 

1002.3^=     4 

20' 

22' 

3536.  4£:  : 

6018.  6 

1003.5             d 

22' 

24' 

3538.7l3 

6021  .4 

IOO4.7           £ 

24' 

26' 

3541.  O.b:  : 

6024  .  2 

1006.  o        *§ 

26' 

28' 

3543-3^ 

6027  .0 

1007.2         Jo, 

28' 

30' 

3545-6  |V 

6029.9 

1008.4          o 

30' 

32' 

3547-9  H-4£ 

6032  .  7 

1009.6            ^ 

32' 

34' 

3550.2  £$° 

6035.6 

1010.8         £ 

34' 

36' 

35.52.5?-  - 

6038.4 

IOI2  .  I            ^g 

36' 

38' 

3554-  8«T  " 

6041  .3 

IOI3.3           < 

38' 

40' 

3557-1 

6044  .  i 

IOI4.5 

40' 

42' 

3559-4 

6047  .0 

IOI5.7 

42' 

44' 

356i.7 

6049  •  8 

1016  .9 

44' 

46' 

3564-0 

6052  .  6 

1018.2 

46' 

48' 

3566.3 

6055.5 

1019  .4 

48' 

50' 

3568.7 

6058.3 

IO2O  .6 

5°,' 

52' 

3571-0 

6o6l  .2 

IO2I  .8 

52 

54' 

3573-3 

6064.0 

1023.0 

54' 

56' 

3575-6 

6066  .  9 

1024.3 

56 

58' 

3577-0             ! 

6069  .  7 

1025.5 

58' 

87 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

64° 

3580.2 

6072.5 

1026.7 

64° 

2' 

3582.6 

6075.3 

1027.9 

2' 

4' 

3584.9 

6078.1 

IO29  .  2 

4' 

6' 

3587.2          4 

6081  .0 

1030.4 

6' 

8' 

3589.5          6 

6083.8 

1031.7 

8' 

10' 

3591-8         * 

6086.6 

1032.9 

10' 

12' 

3594-2         .§ 

6089.4 

I034.I 

12' 

14' 

3596.5         tg1 

6092  .  2 

1035.4                                   14' 

16' 

3598.8          - 

6095  .0 

1036.6 

16' 

18' 

3601.1         ^ 

6097.8 

1037.9 

18' 

20' 

3603.4      *| 

6100.  7 

I039.I             * 

20' 

22' 

3605.7     £^ 

6103.5 

1040.3             6 

22' 

24' 

3608.1      13  ^ 

6106.3 

I04I.6           ^ 

24' 

26' 

3610.4     -g,  ' 

6109.2 

1042  .8        *§ 

26' 

28' 

3612.8     ^ 

6112  .0 

1044.1         '5. 

28' 

o 

30' 

3615-1      X 

6114.8 

1045.3        1 

30' 

32' 

3617.5      % 

6117.6 

1046.5               co 

32' 

34' 

3619.8    .*  + 

6120.4 

1047.8    .  .  % 

34' 

36' 

3622  .1   "??  M 

6123.2 

1049  .0  "*  CT& 

36' 

38' 

3624.5^2 

6126.0 

I05°-3  6.  < 

38' 

40' 

3626.8^     '% 

6128.9 

1051  .5  - 

40' 

42' 

3629.  2  £    £ 

6131.7 

1052.7!: 

42' 

44' 

6i34.5 

1054.  o  & 

44' 

46' 

3633.82     t 

6137.3 

1055.2.0: 

.46' 

48' 

3636.2    M       o 

vo  ^  O 

6140.  i 

1056.5   9  y? 

48' 

50' 

3638.^            '  *~~ 

6143.0 

1057.7           * 

50' 

52' 

3640.9?^| 

6145.8 

1059.  o|:  | 

52' 

54' 

3643.2     -"jp 

6148.6 

IO6O.2                           -H 

54' 

56: 

3645.5         £ 

6151.4 

I06I.5                          .^ 

56' 

58' 

3647.9        | 

6154.2 

IO62.7                          & 

58' 

65° 

3650.2  t; 

6157.0 

1063.9                          <~ 

65° 

2f 

3652.6    *  . 

6159.8 

2' 

4' 

3654.9  ^2 

6162.6 

1066.4 

4' 

6' 

3657-3    3,6 

6165.4 

1067.7                          | 

6' 

8' 

3659.6  ^ 

6168.2 

1068.9 

8' 

10' 

3661.9     1 

6171  .0 

1070.  2 

10' 

12' 

3664.3         cc 

6173.8 

1071  .4 

I2X 

14' 

3666.7      1 

6176.6 

1072.7 

14' 

16' 

3669.0       1? 

6179.4 

1073.9 

16' 

18' 

367L3            o 

6182.2 

1075.2 

18' 

20' 

3673.6     5 

6185.1 

1076.5 

20' 

22' 

3676.0      << 

6187.9 

1077.7 

22' 

24' 

3678.3 

6190  .  7 

1079.0 

24' 

26' 

3680.6 

6193-5 

1080  .  3 

26' 

28' 

3683.0 

6196  .  3 

1081.6 

28'  I 

88 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

65°  30' 

3685-4 

6199.  I 

1082  .9 

65°  30' 

32' 

3687.7 

6201  .9 

1084.2 

32' 

34' 

3690.1     .    .   . 

6204.7 

1085.5 

34' 

36' 

*                     10  O>  ^ 

3692.4 

6207.5 

1086.7 

36' 

38' 

3694-8  o-  s 

6210.3 

1088.0 

38' 

40' 

3697-  2  -g 

6213.2 

1089.3 

40' 

42' 

3699-  5  'PT  : 

6216  .O 

1090  .6 

42' 

44' 

37oi.  9*2 

6218.8 

1091  .9 

44' 

46' 

3704.  2£-  - 

6221  .6 

1093.1 

46' 

48' 

3706.6^1 

6224  .4 

1094.4 

48' 

50' 

3709.0  £«£• 

6227  .  2 

1095.7         * 

50' 

52' 

37II-32:  : 

6230.0 

1097.0         . 

52' 

54' 

3713-7^ 

6232  .8 

1098.3        £ 

54' 

56' 

3716.1 

6235.6 

1099.6        *g 

56' 

58' 

3718.5 

6238.4 

IIOO.Q           "S, 

C/2 

58' 

6O° 

3720.8 

6241  .2 

I  IO2  .  2             £ 

66° 

2' 

3723-2 

6244.0 

H03.5           X 

2' 

4' 

3725-5 

6246.8 

IIO4.8             £ 

4' 

6' 

3727-9          * 

6249.6 

I  I  O6  .O    ^  C"Td 

6' 

8' 

3730-3           c 

6252.3 

II07-3  6.  3 

8' 

10' 

3732.7          * 

6255.1 

iio8.6ST 

10' 

12' 

3735-1        .b 

6257.9 

1109.9  .*?. 

12' 

14' 

3737-5         £ 

6260.7 

I  I  I  I  .  2  £" 

14' 

16' 

3739-8        & 

6263.5 

III2.5|: 

16' 

18' 

3742.2        ^ 

6266.3 

III3  .80^ 

18' 

20' 

3744.6  ^f 

6269  .  I 

«oo 
IIIS-IT-     4 

20' 

22' 

3747-0  6.  -e 

6271  .9 

IIl6   .4    r^;        M 

22' 

24' 

3749.  4^-  < 

6274.7 

1117.  7<    1 

24' 

26' 

375I-7l. 

6277.5 

III9.I                    J 

26' 

28' 

3754.1'J-- 

6280.2 

1120.4       .h 

28' 

30' 

3756.  5|: 

6283.0 

1121.7       ? 

30' 

32' 

3758.9  ^vo    . 

6285.8 

1123.0       ^ 

32' 

34 

3761.3   H^J 

6288.6 

1124.3        £ 

34' 

36' 

3763-7^^ 

6291  .4 

II25-6       ^ 

36' 

38' 

3766.0^:  2 

"^       rt 

6294.1 

1126  .9         »d 

*^ 

38' 

40' 

3768.4           £ 

6296  .9 

1128.2 

40' 

42' 

3770.8           W 

6299.7 

1129.5 

42' 

44' 

3773-2       ,0 

6302.5 

1130.8 

44' 

46' 

3775-6       «? 

6305-3 

1132.2 

46' 

48' 

3778.0        | 

6308.1 

II33-5 

48' 

50' 

3780.4       § 

6310.9 

1134.8 

50' 

52 

3782.8       < 

6313.7 

1136.1 

52' 

54 

3785-2 

6316.5 

H37-4 

54' 

56' 

3787-6 

6319.2 

1138.8 

56' 

58' 

3790.0 

6322.0             II40.I                                   58' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 

Chord. 

External 
Secant. 

Whole     1 
Angle. 

67° 

3792.4 

6324-8 

1141.4 

67° 

2' 

3794-8 

6327.5 

1142.7 

2' 

4' 

3797-2 

6330.3 

1144  .0 

4' 

6' 

3799-6        2 

6333.I 

1145.4 

6' 

8' 

3802  .0          o 

1146.7 

8r 

10' 

3804.4     5 

6338.6 

1148  .0 

10' 

12' 

3806.8      .§ 

6341  .4 

1149.4 

12' 

14' 

3809.2          £ 

6344.2 

1150.7 

14' 

16' 

3811.6     1 

6346.9 

1152  .0 

16' 

18' 

3814.0           a 

6349.7 

IJ53-4 

18' 

20' 

3816.4  •?! 

6352.5 

1154.7  -04 

20' 

22' 

3818.8    £•* 

6355-3 

1156  .0  6. 

22' 

24' 

3821.2      ^< 

6358.1 

1157.4^" 

24' 

26' 

3823.6      £ 

6360.8 

H58.7.1: 

26' 

28' 

3826.0     w 

6363.6 

28' 

30' 

3828.4  t 

6366.3 

Il6l  .4  J5: 

30' 

32' 

3830-9      4 

6369.1 

1162.7    oo 

32' 

34' 

3833-3   :*4 

6371.8 

1164.1    "'* 

34' 

36' 

3835-7  "?SM. 

6374.6 

1*65.4^ 

36' 

38' 

3838.  i  £<£ 

6377-3 

1166  .  7  <T 

38' 

40' 

3840.5^    *g 

6380.1 

ri68.i 

40' 

42' 

3843.0-5,   ;a 

6382.8 

1169  .4 

42' 

44' 

3845.4^  * 

6385.6 

1170.8 

44' 

46' 

3847.8-2  a 

6388.4 

1172.1 

46' 

48' 

3850.2   2"      o 

6391.2 

H73-4 

48' 

50' 

3852.6^  £ 

6394.0 

1174.8 

50' 

52' 

3855.1    TS   ds-tf 

6396.7 

1176  .  i 

52' 

54' 

3857-  5^ 

6399.5 

H77-5 

54' 

56' 

3859-9  £ 

6402  .3 

1178.8 

56' 

58' 

3862.3   | 

6405  .  I 

i  180  .  2 

58' 

68° 

3864.7    <£ 

6407  .9 

1  1  8  1  .  6 

68° 

2' 

3867.1      ,0 

6410.  7 

1183  .  o 

2' 

4' 

3869.6      <^4 

6413-5 

1184.3  ^d4 

4' 

6' 

3872.0       ^c- 

6416  .  2 

1185.7    -      M 

6' 

8' 

3874.4      ^^ 

6419  .O 

1187  .0  ^-  = 

8' 

i  a' 

3876.8      <| 

6421.7 

1188.4^ 

10' 

12' 

3879.3         ^ 

6424.5 

1189.7^  : 

12' 

14' 

3881.7           fc 

6427.2 

II9I    .    I       JH_ 

14' 

16' 

3884.2 

6429.9 

1192    .4    £  \ 

16' 

18' 

3886.6        2 

6432  .6 

1193  .  8   46e  o 

18' 

20' 

3889.0      -| 

6435-4 

1195.2  jgj 

20' 

22' 

3891-5       < 

6438-1 

1196  .  5  <  '  " 

22r 

24' 

3893-9 

6440  .  9 

1197.9 

24' 

26' 

3896.4 

6443-7 

1199.2 

26' 

28' 

3898.8 

6446.4 

1200.6 

28' 

90 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long                 External 
Chord.                 Secant. 

Whole 
Angle. 

68°  30' 

3901.2 

6449.2 

1202  .0 

68°  30' 

32' 

3903-7 

6452.0 

1203.3 

32' 

34' 

3906.1 

6454.7 

1204.7 

34 

36' 

3908.5          4 

6457-5 

I2O6  .  I 

36' 

38' 

3910.8          ^ 

6460  .  2 

1207.5 

38' 

40' 

39I3-4         * 

6463.0 

1208.9 

40' 

42' 

39I5-9          S 

6465.7 

I2I0.3 

42' 

44' 

3918.3       £ 

6468.5 

I2II  .6 

44' 

46' 

3920.7       o£ 

647L3 

1213.0 

46' 

48' 

3923.2       o- 

6474.0 

1214-4      ^4 

48' 

5°' 

3925.6    -g2 

6476.8 

1215.8      d~ 

So' 

5^' 

3928.0    -a^ 

6479.5 

1217.2     ^ 

52' 

54' 

3930.5    <£? 

6482.3 

1218.  6     -3 

54' 

S6' 

3932.9    £ 

6485  .0 

I22O.O       'a: 

56 

58' 

3935-4     7 

6487.8 

1221.4   °2 

58' 

09° 

3937-9     f 

6490  .6 

1222.8       ^ 

69° 

2' 

3940-4     4 

6493.3 

1224.  2       06  a 

2' 

4' 

3942.8    .3  . 

6496.  i 

1225.6     >TJ  H 

4' 

6' 

3945-3  ^  « 

6498.8 

1227.0   "'^ 

6' 

8' 

6501  .6 

1228.4^0 

8' 

10' 

3950.  2  £    ^ 

6504-3 

1229.  8-g 

10' 

12' 

3952.  7  £  •§, 

6507.1 

1231.2-% 

12' 

14' 

3955  -1  £    w 

6509.8 

1232.6^ 

14' 

1  6' 

3957-6^   £ 

6512  .6 

1234.  o£ 

16' 

1  8' 

3960  .  o  M'     »? 

6515  .  3 

I235-4  £ 

18' 

20' 

3962.5^  [£ 

6518.0 

1236.8^0* 

20' 

22' 

3965-0^^, 

6520.8 

1238.23^ 

22' 

24' 

3967.  4     J§< 

6523-5 

1239.6     ^ 

24' 

26' 

3969-9     13 

6526.3 

1241.0     g^ 

26' 

28' 

3972.3     | 

6529.0 

1242.4     J6." 

28' 

30' 

3974-8     *£ 

653I-7 

1243.8     |: 

3°' 

32' 

3977-3     ^4 

6534.5 

1245.2       t^oo 

32' 

34' 

3979-7      *" 

6537-2 

1246.6      °°  ^ 

34' 

36' 

3982.2      £g 

6540  .0 

1248.0     ;g^ 

36' 

38' 

3984.6     3^ 

6542.7 

1249.4     <<" 

38' 

40' 

3987.1     3J, 

6545-4 

1250.8 

40' 

42' 

3989.6        * 

6548.2 

1252  .2 

42' 

44' 

3992-0        £ 

6550-9 

1253.6 

44' 

46' 

3994-5         6 

6553.7 

1255-0 

46' 

48' 

3997-0        £ 

6556-4 

1256.4 

48' 

So; 

3999-5        | 

6559-I 

1257.9 

50' 

4002  .  o 

6561.9 

1259.3 

52' 

54' 

4004  .  5 

6564.6 

1260  .  7 

54 

56' 

4007  .  o 

6567.4 

1262  .  2 

56 

58' 

4009.5            !   6570.1 

1263  .6 

58' 

91 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

70° 

4011  .9 

6572.8 

1265  .0 

70° 

2' 

4014.4 

1266.4 

2' 

4' 

4016.9 

6578^3 

1267.8 

4' 

6' 

4019.4 

6581  .0 

1269.3 

6' 

8' 

4021  .  9 

6583.7 

1270.7 

8' 

IP' 

4024.4         £ 

6586.4 

1272  .  I 

10' 

12' 

4026.9        £ 

6589.1 

1273-5 

12' 

14' 

4029.4        -g 

6591.9 

1275.0 

14; 

16' 

4031.9     *a 

6594.6 

1276  .4 

18' 

4034.4     "2 

6597.3 

1277.9 

18' 

20' 

4036.8     ^ 

6600.0 

1279.3 

20' 

22' 

4039-3      2 

6602  .7 

1280.7 

22' 

24' 

4041.8      *- 

6605.5 

1282  .  2 

24' 

26' 

4044  .  3        ? 

6608.2 

1283.6 

26' 

28' 

4046.8 

6610  .  9 

I285.I 

28' 

30' 

4049  •  3 

6613.6 

1286.5 

30/ 

32' 

4051.8 

6616.3 

1288.0 

327 

34' 

4054.3    .  .  . 

6619  .0 

1289.4 

34' 

36' 

4056.8  ^^M 

6621.8 

1290.8  £f  j 

36' 

38' 

4059-3;°:  s 

6624.5 

1292.3  6:  : 

38' 

40' 

4061  .8  *3 

6627  .  2 

1293  .  7  ^ 

40' 

42' 

4064.  4  •£:  = 

6629  .  9 

1295.  2-  £:  : 

42' 

44' 

4066.9^ 

6632  .6 

1296  .6  cc 

44' 

46' 

4069.  4.  2:  = 

6635.4 

1298  .  I  J3:  s 

46' 

48' 

4071  .9  ^  ^ 

6638.1 

1299.5  £g  % 

48' 

50' 

4074.45^ 

6640.8 

1301.0^^ 

50' 

52' 

4076.9?:  - 

6643.5 

52' 

54' 

4079.  4^ 

6646.2 

1303-9^ 

54' 

56' 

4081  .9 

6649  .0 

1305.3 

56' 

58' 

4084.4 

6651.7 

1306.8 

58' 

71° 

4086.9 

6654.4 

1308.3 

71° 

2' 

4089.5         j 

6657.1 

1309.7 

2' 

4' 

4O92      .O                                0 

6659.9 

I3II  .  2 

4' 

6' 

4094.5         52 

6662.6 

I3I2.6 

6' 

8' 

4097.0        | 

6665.3 

I3I4.I 

8' 

10' 

4099.5         £ 

6668.0 

I3I5.6 

10' 

12' 

4102  .0          o 

6670.  7 

I3I7.0 

12' 

14' 

4104.6         o 

6673.4 

I3I8.5 

14' 

16' 

4107.0          2 

6676.1 

1320.0 

16' 

18' 

4109.5         £ 

6678.8 

I32I.4 

18' 

20' 

4112  .  I         << 

6681.5 

1322  .9 

20' 

22' 

4114.6 

6684.2 

1324.4 

22' 

24' 

4117.2 

6686.9 

1325.8 

24' 

26' 

4119.7 

6689.6 

I327.3 

26' 

28' 

4122  .  2 

6692.3   '     1328.8                      28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

71°  30' 

4124.7 

6695.0 

1330.3 

71°  30' 

32' 

4127.3 

6697.7 

I33L8 

32/ 

34' 

4129  .8 

6700  .4 

1333.2 

34' 

36' 

4132.4  ^4 

6703.1 

1334.7        •    • 

36' 

38' 

4134.9  6     : 

6705.8 

1336.2  +** 

38' 

40' 

4137.4  fc" 

6708.5 

1337.7  ^:  : 

40' 

42' 

4140.0  £^ 

6711  .2 

1339  .  2  ^ 

42' 

44' 

4142.5  '&•  : 

67I3-9 

1340.  6  •£=  : 

44' 

46' 

4I45-0   fcu  , 

6716.6 

1342  -IC£L  . 

46' 

48' 

4147  .6  £"M"a 

6719.3 

48' 

5o; 

4150.  i  ££2 

6722  .O 

H  00    O 

I345.I   ^odo- 

5o' 

4152.7  ^* 

6724.7 

'I346.6,d 

52,' 

54' 

4155-2  5=  = 

6727.4 

1348.13=  = 

54' 

56' 

4157-8 

6730.1 

1349.6 

56 

58' 

4160.3 

6732.8 

I35I  -1 

58' 

72° 

4162  .8 

6735.5 

1352.6 

72° 

2' 

4165.4 

6738.2 

I354.I 

2' 

4' 

4167.9 

6740.9 

1355-6 

4' 

6' 

4170.5         * 

6743.6 

6' 

8' 

6746.3 

1358*6 

8' 

10' 

4I75-6 

6749.0 

1360.1 

10' 

12' 

4178.1        £ 

675J.7 

1361.6 

ia' 

14' 

4180.7        eg 

6754.4 

1363-1 

14' 

16' 

4183.2        j> 

6757-1 

1364.6 

16' 

18' 

4185.8           o 

6759.8 

1366.1 

18' 

20' 

4188.4   <oa£. 

6762.5 

1367.6  ^j 

20' 

22' 

4I9LO  |:  5 

6765.1 

I369.I      O'.     ; 

22' 

24' 

4193.5  -3     < 

6767.8 

1370.65 

24' 

26' 

4196.1  .53. 

6770.5 

1372.2^.     . 

26' 

28' 

4198.6  & 

6773.2 

J373-7C& 

28' 

30' 

4201  .2   £: 

6775.9 

1375.2    0=    5 

30; 

32' 

4203.8    J^4 

6778.6 

1376.7    ^^^ 

32 

34' 

4206.3    jf^- 

6781.3 

1378.  2^MS 

34 

36' 

4208.9    TJ       j° 

6784.0 

36 

38' 

4211     .4     ^       ,-H 

6786.7 

1381.  3<f  " 

38' 

40' 

4214.0      -a 

6789.3 

•1382.8 

40' 

42' 

4216.6      ^ 

6792  .0 

1384.3 

42' 

44' 

4219.1      £ 

6794.7 

1385-8 

44' 

46' 

4221.7      ^ 

6797.4 

1387-4 

46' 

.48' 

4224.2      - 

6800.  i 

1388.9 

48' 

50' 

4226.8      ? 

6802.7 

1390.4 

50' 

52' 

4229.4 

6805  .4 

i39i-9 

52' 

54' 

4231.9 

6808.0 

1393.4 

56' 

4234.5 

6810.7 

i395.o 

56' 

58' 

4237-1 

6813.4 

1396-5 

58' 

93 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

73° 

4239.7 

6816.  I 

1398.0 

73° 

2' 

4242.3 

6818.  8 

1399  .  5 

2' 

4' 

4244.8 

6821.5 

1401  .  i 

4' 

6' 

4247.4 

6824.2 

1402  .6 

6' 

8' 

4250.0 

6826.8 

1404.  2 

8'' 

10' 

4252.6 

6829.5 

1405.7 

10' 

12' 

4255  .  2 

6832.2 

1407-3 

12' 

14' 

4257  «  7 

6834.9 

1408.8 

14' 

iff 

4260.3 

6837-5 

1410  .4 

16' 

18' 

4262  .9 

6840.2 

1411.9 

18' 

20' 

4265.5      4 

6842.8 

1413.5  ^  .  . 

20' 

22' 

4268.1 

6845.5 

1415  .0  >0<>« 

22' 

24' 

4270.7      £ 

6848.2 

1416.  6£:  : 

24' 

26' 

4273.3     i 

6850  .  9 

1418.1- 

26' 

28' 

4275-9     -~ 

6853.5 

1419  -7  -a-  : 

28' 

30' 

4278.5    <£ 

6856.2 

1421.  2  <£ 

30' 

32' 

4281.1       £ 

6858.9 

1422  .8  <-2:  - 

32' 

34' 

4283.7   .  .;? 

6861  .6 

1424.3  £J'J 

34' 

36' 

4286.3  ^a^ 

6864.3 

1425  .9         N 

36' 

38' 

4288.  9  kdr  ;§ 

6866.9 

1427.  4|:  : 

38' 

40' 

4291.5-3    ^ 

6869.6 

1429.0 

40' 

42' 

4294.1  .£: 

6872.3 

1430.6 

42' 

44' 

4296.702 

6874.9 

1432.1 

44' 

46' 

4299.3  ,0  = 

6877.6 

1433-7 

46' 

48' 

4301  .9   <?£4 

6880.2 

1435-2 

48' 

5°' 

4304.5  ss: 

6882.9 

1436.8 

50' 

52/ 

4307.1  ?:  fc 

6885.5 

1438.4 

52' 

54' 

4309.7^     *g 

6888.2 

1439-9 

54' 

56' 

4312.3      -a 

6890  .  9 

I44L5 

56' 

58' 

4314.9     <£ 

6893.6 

1443.0 

58' 

74° 

4317-6    i 

6896.3 

1444.6 

74° 

2' 

4320.2     « 

6899  .0 

1446.2 

2' 

4' 

4322.9    j 

6901  .6 

1447.8  ^°? 

4' 

6' 

4325-5 

6904.3 

1449-3,°:  : 

L    6' 

8' 

4328.1 

6906  .  9 

14  CQ  .  9  ^ 

"15 

r"p  s' 

10' 

4330.7 

6909.6 

JH  ^      „ 

ior 

12' 

4333-4 

6912  .  2 

1454.1  <£ 

12' 

14' 

4336.0 

6914.8 

1455-  7^:  2 

?j    14' 

16' 

4338.6 

6917.5 

1457.2   «  9  t 

16' 

18' 

4341.2 

6920.  I 

1458.  8^a« 

18' 

20' 

4343-8 

6922  .8 

1460.4^-  = 

20' 

22' 

4346.5 

6925-5 

1462  .0 

22' 

24' 

4349  -1 

6928.2 

1463.6 

24' 

26' 

435*-7 

6930.8 

1465.2 

26' 

28' 

4354-3 

6933-5 

1466.8 

28' 

94 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

74°  30' 

4356-9 

6936.1 

1468  .4 

74°  30' 

32' 

4359-6 

6938.8 

1470  .0 

S^' 

34' 

4362.2 

6941.5 

1471.6 

34' 

36' 

4364.9  ,A^4 

6944.1 

1473.2    . 

36' 

3*' 

4367-5  c.  ,M 

6946.8 

1474-8  "*2 

38' 

4C/ 

4370-1  S" 

6949.4 

1476.4;!"  = 

40' 

42' 

4372.8  g.  . 

6952  .O 

1478.0-3 

42' 

44' 

4375  -4  c&"  " 

6954.7 

1479.  6  -gr  : 

44' 

46' 

4378.1  £;  - 

6957-3 

1481.2  <« 

46' 

48' 

4380.7^,^ 

6960  .O 

1482.  8  £=.= 

48' 

50' 

4383-3  jj££ 

6962  .6 

<N  q  -<t 
1484  -4  ^  Ov  o 

So' 

52' 

4386.0^ 

6965  .  2 

1486.0  T- 

52' 

54' 

4388.6^  = 

6967.9 

1487.6^=   = 

54' 

56 

4391.2 

6970.5 

1489  .  2 

56' 

58' 

4393-9 

6973-2 

1490.8 

58' 

75° 

4396.5 

6975.9 

1492.4 

75° 

2' 

4399-2 

6978.6 

1494.0 

2' 

4' 

4401-8         4 

6981.2 

1495.6 

4' 

6' 

4404.5  •      - 

6983.9 

1497-3 

6' 

8' 

4407.1        ,c 

6986.5 

1498.9 

8' 

10' 

4409.8        -g 

6989.  I 

J5oo.5 

id* 

12' 

4412.5        •£ 

6991  .8 

1502  .  i 

12' 

14' 

44i5.i        ? 

6994.4 

I5°3-7 

14' 

16' 

4417-8        ^ 

6997.0 

1505-4 

1  6' 

18' 

4420.5         2" 

6999-7 

1507.0 

18' 

20' 

4423.1  ^^^ 

7002.3 

1508.6  ^»  j 

20' 

22' 

4425.8^:  | 

7005  .0 

1510.2  6-  . 

22' 

24' 

4428.4-    < 

7007  .6 

1511.8^^  " 

24' 

26' 

443i.i-a: 

7010.2 

J5iS.sl:  s 

26' 

28' 

4433-8^ 

7012  .8 

ISIS-  leg1"  " 

28' 

3°! 

4436.  4  -2: 

7015-4 

1516.  7|:  = 

30' 

32' 

4439.1  ^4 

7018.0 

1518.3   „« 

32' 

3£ 

4441  .7  '^^^ 

7020.7 

1520.0  roaS 

34' 

36 

4444-  4  ?-  fc 

7023.3 

1521.65.  . 

36' 

38' 

4447-  1  <"  *g 

7026  .0 

i523-3<"  " 

38' 

40' 

4449  •  7        £ 

7028.6 

1524-9 

40' 

42' 

4452.4          £ 

7031.2 

1526.5 

42' 

44' 

4455-0        £ 

7033.9 

1528  .  2 

44' 

46' 

4457-7         : 

7036.5 

1529.8 

:    46' 

48' 

4460  .4         ^ 

7039.2 

!53i-5 

;  48' 

5°; 

4463.1        | 

7041  .8 

i533.i 

s°: 

52' 

4465.8 

7044.4 

J534.7 

52' 

s£- 

4468  .4 

7047.1 

1536.4 

54' 

5o 

4471  .1 

7049.7 

1538-0 

56' 

58' 

4473-8 

7052.3 

T539-7 

'  58' 

95 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

76° 

4476.5 

7055-0 

I54L4 

76° 

2' 

4479.2 

7057-6 

1543-0 

2' 

4' 

4481  .8 

7060  .  2 

1544.7 

4' 

6' 

4484.5 

7062.8 

1546.3 

6' 

8' 

4487.1 

7065.4 

1548  .0 

8' 

10' 

4489.9 

7068.1 

1549-7 

10' 

12' 

4492  .6 

7070.7 

I55I-3 

12' 

14' 

4495-3 

7073-3 

1553-0 

14' 

16' 

4498  .0 

7075-9 

1554-7 

16' 

18' 

4500.7 

7078.5 

1556.3 

18' 

20' 

4503-4         4 

7081  .  2 

1558.0 

20' 

22' 

4506.1 

7083.8 

J559-7 

22' 

24' 

4508.8        1 

7086  .4 

1561.3 

24' 

26' 

45I]C-4        - 

7089.0 

1563-0 

26' 

28' 

45i4.i        £ 

7091  .6 

1564-7 

28' 

30' 

4516.9      <£ 

7094.2 

1566.4 

30' 

32' 

4519.6      a 

7096  .8 

1568.1 

32' 

34' 

4522.3      U 

7099.4 

1569  .  8 

34' 

36' 

4525.0  ^4^ 

7102  .0 

1571.4^? 

36 

38' 

4527'  7  £:  '5 

7104.6 

1573-  l|:  = 

38' 

40' 
42' 

4530.4^    ^ 
4533.  2  •-  = 

7107.3 
7109.9 

1574-  8  -g 

1576.  5  ft:  : 

40' 
42' 

44' 

4535-9^ 

7112.5 

1578.2^ 

44' 

46' 

4538.6^: 

7115.1 

1579.8^=  5 

46' 

48' 

4541-3  3^<4 

7117.7 

^81.5  SSI- 

48' 

5o; 

4544.0  £3" 

7120.4 

1583.2^ 

5°,' 

4546.8^  & 

7123.0 

1584-  9S:  : 

52' 

54' 

4549-  5  <    "g 

7125.6 

1586.6 

54 

56' 

4552.2        'a 

7128.2 

1588.2 

56 

58' 

4554-9        ^ 

7130.8 

1589.9 

58' 

77° 

4557-6        ^ 

7133.5 

1591.6 

77° 

2' 

4560.3         « 

7136.1 

1593-3 

2' 

4' 

4563.0        *- 

7138.7 

1595-0 

4' 

6' 

4565.7        •§ 

7I4L3 

1596.7 

6' 

8' 

4568.4 

7143.9 

1598.4 

8' 

10' 

4571-2 

7146.5 

1600.  I 

10' 

12' 

4573-9 

7149.1 

1601.8 

12' 

14' 

4576.6 

7151  .  7 

1603.5 

14' 

16' 

4579-3 

7T54.3 

1605  .  2 

16' 

18' 

4582  .0 

l6o6  .9 

18' 

20' 

4584.8 

7159.5 

1608.  6 

20' 

22' 

4587.5 

7162  .  i 

1610.3 

22' 

24' 

4590.3 

7164.7 

1612  .0 

24' 

26' 

4593-0 

7J67-3 

1613.7 

26' 

28' 

4595-7 

7169.9 

1615.4 

28' 

96 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

77°  30' 

4598.5 

7*72.5 

1617  .  I 

77°  30' 

32' 

4601  .  2 

7i75.i 

1618.8 

32' 

34' 

4604.0 

7177.7 

1620.5 

34' 

36' 

4606.7  ^^4 

7180.3 

l622  .  2 

36' 

38' 

4609.5  6_  j 

7482.9 

1624.0 

38' 

40' 

4612  .  2  J" 

7185.5 

1625.7 

40' 

42' 

4615.0!^  . 

7188.1 

1627.5 

42' 

44' 

46l7.7l"    " 

7190.7 

1629  .  2 

44' 

46' 

4620.5    ^    : 

7I93-3 

1630.9 

46' 

48' 

4623.3^; 

7195.9 

1632.7 

48' 

50' 

4626    .O      £>  £   H 

7198.5 

1634.4 

50' 

S^' 

4628.8^* 

7201  .  i 

1636.1 

52' 

54' 

4631-  5  3j:  : 

7203.7 

1637.9 

54' 

56' 

4634.2 

7206.3 

1639.6 

56' 

58' 

4637.0 

7208.9 

1641.3 

38' 

78° 

4639-7 

7211.5 

1643.0 

78° 

2' 

4642  .5 

7214.1 

1644  .8 

2' 

4' 

4645.2 

7216.7 

1646.5  ^.^ 

4' 

6' 

4648  .0 

7219.3 

1648.3    t     - 

6' 

8' 

4650,8 

7221.9 

1650.0  j°:  : 

8' 

10' 

4653-5 

7224.4 

1651-7   g 

10' 

12' 

4656.3 

7227.0 

1653.  5  -a:  : 

12' 

14' 

4659.0 

7229.6 

1655.2^ 

14' 

16' 

4661.8 

7232.2 

1657.  o^  = 

16' 

18' 

4664.6 

7234-8 

'658.7;-  So 

18' 

20' 

~*          iA  <>  4 
4667.4         « 

7237-3 

1660.5  -a 

20' 

22' 

4670.2^:  : 

7239-9 

1662.  3^^  : 

22' 

24' 

4673.0-3 

7242.5 

1664  .  o 

24' 

26' 

4675.7^  : 

7245.0 

1665.8 

26' 

28' 

4678  .  5  c/2 

7247.6 

1667.5 

28' 

3°; 

4681.  3  I55 

7250.2 

1669.3 

30' 

32' 

4684.1  ^J 

7252.8 

1671  .0 

32' 

34' 

4686.9  £3;: 

7255.4 

1672.8 

34' 

36' 

4689.6;^  „ 

7258.0 

1674.5 

36' 

38' 

4692  .4  <j" 

7260.6 

1676.3 

38' 

40' 

4695.2 

7263.2 

1678.  I 

40' 

42' 

4698.0 

7265.8 

1679.9 

42' 

44' 

4700.8 

7268.3 

1681.7 

44' 

46' 

4703.6 

7270.9 

1683.4 

46' 

48' 

4706.4 

7273-5 

1685.2 

48' 

5°' 

4709  .  2 

7276.1 

1687.0 

5°' 

5^' 

4712.0 

7278.7 

1688.8 

52' 

54' 

4714.8 

7281.3 

1690  .6 

54' 

56' 

4717.6 

7283.9 

1692.3 

56' 

58' 

4720.4 

7286.5 

1694.  i 

58' 

97 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Chord. 

External 
Secant. 

Whole 
Angle. 

79° 

4723-1 

7289  .  0 

1695.9 

79° 

2' 

4725-9 

7291  .  6 

1697.7 

2' 

4' 

4728.7 

7294.1 

1699.5 

4' 

6' 

4731-5 

7296.7 

1701.3 

6' 

8' 

4734-3 

7299.2 

1703.0 

8' 

10' 

4737-1 

7301.8 

1704  .  8 

10' 

12' 

4739-9 

7304.3 

1706  .  6 

I2f 

14' 

4742.7 

7306.9 

1708  .4 

14' 

16' 

4745-5 

7309-5 

I7IO  .  2 

16' 

18' 

4748.3 

7312.0 

1712  .0 

18' 

20' 

4751-2 

7314.6 

1713.7     .    .    . 

20' 

22' 

4754.0 

7317  .  2 

I7I5-5    ^C'- 

22^ 

24' 

4756.9        £ 

73J9-8 

24' 

26' 

4759-7        ^ 

7322  .  3 

1719.1^ 

26' 

28' 

4762.5        .fa 

7324.9 

I72O  .9  ,|H:     , 

28' 

3°' 

4765.3        <£ 

7327.4 

I722.7f 

30' 

32' 

4768.2        £ 

733o.o 

1724.5,^ 

32' 

34' 

4771  .0        ^ 

7332.5 

34' 

36' 

4773.8  lAor- 

7335'  J 

I728.I      "'*« 

•36' 

38' 

4776.6  6,  ^ 

7337-6 

1729.  9|:  - 

38' 

40' 

4779-4-a'    ^ 

-9340.2 

I73L7 

40' 

42' 

4782.  2-g: 

7342.8 

1733.5 

42' 

44' 

4785-  IC£ 

7345-3 

1735.3 

44' 

46' 

4787.  9& 

7347-9 

I737-1 

46' 

48' 

4790.7  224 

7350.4 

1738.9 

48' 

50' 

4793.6  S3J: 

7353-0 

1740.8 

50' 

52' 

4796  .4  ^g,  £ 

7355-5 

1742.6 

52' 

54' 

4799.  2<T  | 

7358.1 

1744.4 

54' 

56' 

4802.0         '£, 

736o.6 

1746.2 

56' 

•     58' 

4804.9         ^ 

7363-2 

1748.0 

58' 

8O° 

4807.7         f. 

7365.8 

1749.9 

8O° 

2' 

4810.5          N 

7368.4 

I75L7  u.<^4 

2' 

4' 

4813.3     5 

7370.9 

4f 

6' 

4816.2      5 

7373-5 

1755.4  6.  i 

6' 

8' 

4819.0 

7376.0 

1757.2:: 

8' 

10' 

4821  .9 

7378.5 

1759.  o|:  : 

10' 

12' 

4824.7 

7381.1 

1760  .  8  M 

12' 

14' 

4827.6 

7383-6 

1762.6,0:  : 

14' 

16' 

4830.4 

7386.2 

1764.5  4"?^ 

16' 

18' 

4833-3 

7388.7 

1766.3  "°^ 

18' 

20' 

4836.2 

7391.2 

1768.  2|=  : 

20' 

22' 

4839.0 

7393-8 

1770.0 

22' 

24' 

4841.9 

7396.3 

1771.9 

24' 

26' 

4844.8 

7398.9 

J773-7 

26' 

28' 

4847.6 

74oi  .4 

1775.6 

28' 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

80°  30' 

4850.5 

7404.0 

1777-4 

80°  30' 

32' 

4853  .4 

7406.5 

1779-3 

32' 

34' 

4856.2 

7409.1 

1781.1 

34' 

36' 

4859.1  ^^4 

74H.6 

1783.0 

36' 

38' 

4862.0    . 

7414.2 

1784.8 

38' 

40' 

4864.  8  £:  : 

7416.8 

1786.7 

40' 

42' 

4867.7*8.  , 

74I9-3 

1788.5 

42' 

44' 

4870.6$-  ' 

#42  I.  9 

1790.4 

44' 

46' 

4873.4  £.  : 

7424.4 

1792.2 

46' 

48' 

4876.3^  a 

7427.0 

1794.1 

48' 

5°' 

4879.2  £^S 

7429.5 

1796.0          ^ 

50' 

52' 

4882.0^ 

7432.1 

1797.9                      « 

5*' 

54' 

4884.  9  2=  : 

7434-6 

1799.7                    g 

CA 

56' 

4887.8 

7437-2 

1801.6        - 

56' 

58' 

4890.7 

7439-7 

1803.5        .£ 

58' 

81° 

4893.6 

7442.2 

1805.4     <£ 

81° 

2' 

4896.5 

7444-7 

1807.3     & 

2' 

4' 

4899.3 

7447-2 

1809.1 

4' 

6' 

49O2  .  2 

7449-8 

1811  .0    .  .  * 

6' 

8' 

4905.1 

7452.3 

1812.9  ^| 

87 

10' 

4907.9 

7454-8 

1814.  8^T 

10' 

12' 

4910.8 

7457-4 

1816.7  $ 

12' 

14' 

4913.7 

7459-9 

1.818,5$- 

4' 

16' 

4916.6 

7462.4 

1820.4  g- 

1  6' 

18' 

49190 

7464.9 

1822.3  ^4 

1  8' 

20' 

4922.4   in0? 

7467.5 

1824.2  ^^ 

20' 

22' 

4925-35=    = 

7470.0 

1826.0  2'  ^ 

227 

24' 

4928.2* 

7472.5 

1827.9^     -g 

24/ 

26' 

4931-*.*:  : 

7475-1 

1829.8      -a 

26' 

28' 

4934.0  c& 

7477-6 

1831.7      \ 

28' 

30' 

4936.9^=  = 

7480.1 

1833.6      t 

30; 

32' 

4939.8  555 

7482.6 

1835.5      i 

32 

34'  1 

4942.7  £$£ 

7485.1 

1837.4      3 

34 

36' 

4945-6;d 

7487.7 

J839-3        < 

36 

38' 

4948.5?:  : 

7490.2 

1841  .2 

38' 

40' 

4951  -4 

7492.7 

1843.1 

40' 

42' 

4954-3 

7495-2 

1845  .0 

42; 

44' 

4957-2 

7497-7 

1846  .9 

44 

46' 

4960  .  2 

7500.2 

1848.8 

46' 

48' 

4963.1 

7502.7 

1850.7 

48' 

50' 

4966  .  o 

7505.3 

1852.6 

5°; 

52' 

4968.9 

7507.9 

1854.5 

52 

54' 

497J-9 

75*0.4 

1856.4 

54 

56' 

4974-8 

7512   9 

1858.3 

56 

58' 

4977-7 

75*5-4 

1860.2 

58' 

99 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

82° 

4980.6 

75J7-9 

1862.2 

82° 

2' 

4983.6 

7520.4 

1864.1 

2' 

4' 

4986.5 

7522.9 

1866.0 

4' 

6' 

4989.4 

1867.9 

6' 

8' 

4992.4 

7527-9 

1869.8 

8' 

10' 

4995-3 

7530.4 

1871.8 

10' 

12' 

4998.3 

7532.9 

1873.7 

I2r 

14' 

5OOI  .  2 

7535-4 

1875.6 

14' 

16' 

5004.2 

7537-9 

1877.6 

16' 

18' 

5007.1 

7540.4 

1879-5 

i8r 

20' 

5010  .0  ^  a  4 

7543-o 

1881  .5 

20' 

22' 

5OI3-o  £  i 

7545-5 

1883.4         4 

22A 

24' 

5015.95-  - 

7548.0 

1885.3         6 

24' 

26' 

5018.9  'g 

7550.5 

1887.3        fc 

26' 

28' 

5021.8'Br  : 

0                     C/3 

7553-o 

1889.2        *g 

28' 

30' 

5024.7  £-  - 

7555-5 

1891  .  2            ,£ 

30' 

32' 

5027.  7  *£„"„ 

7558  .0 

1893.1              & 

32' 

34' 

5030.6  ««« 

7560.5 

1895.1           o 

34' 

36' 

5033-6^^- 

l897    -°                    N 

36' 

38' 

5036.55:  : 

7565.5 

1899.0    ^C/g 

38' 

40' 

5039.5 

7568.0 

1900.9  ^  **> 

40' 

42' 

5042.4 

7570-5 

1902.9-3 

42' 

44' 

5045-4 

7573-o 

1904.8-^: 

44' 

46' 

5048.4 

7575-5 

1906.8^ 

46' 

48' 

5051  -3 

7578.o 

1908.  7  £• 

48' 

5°' 

5054-3 

7580.6 

I9I0.7     2ln 

50/ 

52' 

5057.3 

7583.1 

1912.63.  h° 

52' 

54' 

5060  .  2 

7585.6 

1914  .6  <J"  ^ 

54' 

56' 

5063.2 

7588.1 

1916.5        I 

56' 

58' 

5066  .  I 

7590.6 

1918.5       £ 

58' 

83° 

5069  .  I 

7593-1 

1920.5           £ 

83° 

2' 

5072.1  ^4 

7595-6 

1922.5      T 

2' 

4' 

5075.1  c- 

7598.1 

1924.4 

4' 

6' 

5078.  o£:  = 

7600.5 

1926.4       5 

6' 

8' 

5081  .0-3 

7603  .  o 

1928.4        <- 

8' 

10' 

5084.  og"  : 

7605.5 

1930.4 

10' 

12' 

5087  .0  g. 

7608  .0 

1932.4 

12' 

14' 

5089.  9  xVo 

7610.5 

1934.4 

14' 

16' 

5092.9     £VC^ 

7613.0 

1936.3 

16' 

18' 

5095.9^^ 

7615.5 

1938.3 

1  8' 

20' 

5098.  93j:  : 

7618.0 

1940.3 

20' 

22' 

5101.9 

7620.5 

1942.3 

22' 

24' 

5104.9 

7623.0 

1944.3 

24' 

26' 

5107.8 

7625.5 

1946.3 

26' 

28' 

5110.8 

7628.0 

1948.3 

28' 

100 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord  . 

External 
Secant. 

Whole 
Angle. 

83°  30' 

51*3-8 

7630.4 

1950.3 

83°  30/ 

3*; 

5116.8 

7632.9 

1952.3 

32' 

34' 

5119.8 

7635  •  4 

1954.3 

34' 

36' 

5122.8 

7637.9 

1956.3 

36' 

38' 

5*25.8 

7640.4 

1958.3 

40' 

5128.8 

7642.9 

1960.3 

.4or 

42' 

5*3*-9 

7645.4 

1962.3 

42' 

44' 

5*34-9 

7647.8 

1964.3 

44/, 

46' 

5*37-9 

7650.3 

1966.3 

48' 

5140.9 

7652.8 

1968.3 

48' 

So; 

5*43-9        4 

7655.3 

I970-3 

50/ 

5146.9 

7657.8 

1972.3                      H 

*2' 

54' 

5*49-9        j§ 

7660.3 

1974.4          6 

54' 

56' 

5*52-9        -3 

7662.8 

1976.4         ^ 

56' 

58' 

5*55-9        £ 

7665.3 

1978.4          g 

58' 

84° 

5*58.9     <£ 

7667.8 

1980.4         </? 

84° 

2' 

5161.9       £ 

7670.2 

1982.4         j: 

2' 

4' 

5165.0   .  .5 

7672.7 

1984.5           <? 

4' 

6' 

5168  .0  ^  ^ 

7675.2 

1986.5         5 

6' 

8' 

5*7*-o£:  5 

7677.6 

1988.5  ^5 

8' 

10' 

5*74.015    ^ 

7680.1 

1990.  5|-  ^ 

10' 

12' 

5*77-*-a: 

7682.6 

12' 

14' 

5180.1  co 

7685.1 

1994.  6.  |: 

*4' 

16' 

5*83-  *Ja: 

7687.6 

1996.6  tt 

1  6' 

18' 

5186.2  jrt4 

7690.0 

1998.6,0: 

18' 

201 

5189.2  «  sr 

7692.5 

2000.  7  ^  £H 

20' 

22' 

5I92.2:g:    £ 

7695.0 

2002  .  7  -d   'jg 

22' 

24' 

5*95-3^    *g 

7697  •  5 

2004.  7  ^:  ^ 

24' 

26' 

5*98.3        £ 

7699.9 

2006.8      .b 

26' 

28' 

5201.4        b 

7702.4 

2008.8         c& 

28' 

30' 

5204.4       '~ 

7704.8 

2010.9       £ 

3°' 

32' 

5207.4        ^ 

7707-3 

2012.9        ° 

32' 

34' 

5210.5        T^- 

7709.8 

2015.0      ^ 

3< 

36' 

5213-5      3 

77*2-3 

2017.0       ^ 

38' 

5216.6 

7714.8 

2019  .  I 

38' 

40' 

5219.6 

77*7-2 

2O2I  .  I 

40' 

42' 

5222  .  6 

7719.7 

2023.2 

42' 

44' 

5225.7 

7722.1 

2O25  .  2 

44' 

46' 

5228.8 

7724.6 

2027.3 

46' 

48' 

5231-9 

7727.0 

2029.3 

48' 

50' 

5234.9 

7729  5 

2031.4 

50' 

52' 

5238.0 

7732.0 

2033.4 

52' 

54' 

5241  .0 

7734-4 

2035-5 

54 

56' 

5244.1 

7736.9 

2037.6 

56 

58'       5247.2 

7739-3 

2039.6 

58' 

101 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

85° 

5250.2 

7741-8 

2041.7 

85° 

2' 

5253  .  3 

7744.2 

2043-7 

2' 

4' 

5256  .4 

7746.7 

2045  -8 

4' 

6' 

5259  •  5           "t 

7749-1 

2047.9 

6' 

8' 

5262  .6 

7751-6 

2O5O.O                          H 

8' 

10' 

5265.6         ^ 

7754.1 

2O52     .     I                        £ 

10' 

12' 

5268.7         | 

7756.5 

2054.1                        ^ 

12' 

14' 

5271.8        £ 

7759.0 

2056.2                        £ 

14' 

16' 

5274.9        £ 

7761.4 

2058.3                        * 

16' 

18' 

5278.0       - 

7763-9 

2060.4                       £ 

1  8' 

20' 

5281.0  ^  c.  j 

7766.4 

2062     .5 

20'- 

22' 

5284.1  £^ 

7768.8 

2064.6       ^^ 

22' 

24' 

5287.2*-  ^ 

7771  .3 

2066  .7  -;    3 

24' 

26' 

5290.3  g 

7773-7 

2068.  8g: 

26' 

28' 

5293.4^: 

7776.1 

2070.9  % 

28' 

30' 

5296.4  c: 

7778.6 

2073  .0  c& 

30' 

32' 

5299.5^ 

7781.0 

2075  .  I   o: 

32' 

34' 

5302.6  ««2- 

7783.5 

2O77  .  2  ^-oo  * 

34' 

36' 

5305.7^*0 

7785.9 

2079.3^6 

36' 

38' 

5308.8^=  fc 

7788.4 

2081  .4  13:  ^ 

38' 

40' 

5311.9    4 

7790.8 

2083.5         -| 

40' 

42' 

5315.0         cc 

7793-2 

2085.6         co 

42' 

44' 

5318.1          ,0 

7795-7 

2087.7        J5 

44' 

46' 

5321.2      fl 

7798.1 

2089.9         « 

46' 

48' 

5324.3      J 

7800.6 

2092.0         S 

48' 

50' 

5327.4     :§ 

7803.1 

2094.  i 

50' 

52' 

5330.5     < 

7805.5 

2096  .  2 

52' 

54' 

5333.6 

7808.0 

2098.3 

54' 

56' 

5336.7 

7810.4 

2100.5 

56' 

58' 

5339.8 

7812.8 

2IO2  .6 

58' 

86° 

5342.9 

7815-3 

2104.7 

86° 

2' 

5346.0 

7817-7 

2106.  8    .  ^  . 

2' 

4' 

5349.1  xAd4 

7820.  2 

2109  .0  ^  **» 

4' 

6' 

5352.  2  ^  ^ 

7822.6 

2III  .26.. 

6' 

8' 

7825.0 

2113-3^  " 

8' 

10' 

5358.51,  : 

7827.4 

2115  .4  -^:    = 

10' 

12' 

536i  .6  ^ 

7829.8 

2117  .  5  w 

12' 

14' 

5364.8    fc-    - 

7832.3 

2119  .  7  £:  - 

14' 

16' 

5367.9Xo^ 

7834.7 

2  121  .8    ^^^ 

16' 

18' 

5371.0  m 

7837.1 

2124.0    ^^^ 

18' 

20' 

5374.2^ 

7839-5 

2126.  I  J:    : 

20' 

22' 

5377-3<:  : 

7842  .O 

2128.2 

22' 

24' 

5380.5 

7844.4 

2130.4 

24' 

26' 

5383-6 

7846.8 

2132.5 

26' 

28' 

5386.8 

7849.  2 

2T34-7 

28' 

102 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle  . 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

\Vhole 
Angle. 

86°  30' 

5389-9 

7851.6 

2136.9 

86°  30' 

32' 

5393-0 

7854.0 

2139.0 

32/ 

34' 

5396.2  ^4 

7856.5 

2141  .  2 

34' 

36' 

5399-3    . 

7858.9 

2143-3 

36' 

38' 

5402.5*°:  : 

7861.4 

2145-5 

38' 

40' 

5405.  6  ^  _ 

7863.8 

2147-7 

40' 

42' 

5408.  8  ;a-  - 

7866.3 

2149.8 

42' 

44' 

5412.0  ^ 

7868.7 

2152.0 

44' 

46' 

5415  .  2  <"" 

7871.1 

2I54.I 

46' 

48' 

5418.3  ||  4 

7873.5 

2156.3 

48' 

50' 

5421  .4  "  ^ 

7875  -9 

2158.5 

5°' 

52' 

5424.6?:   = 

7878.3 

2160.6             4 

52' 

54' 

5427.7 

7880.8 

2162.8             ^ 

54' 

56' 

5430.8 

7883.2 

2165.0           X 

56' 

58' 

5434.0 

7885.6 

2167.2            *g 

58' 

87° 

5437-2 

7888.0 

2169.4           $ 

87° 

2' 

5440.4 

7890.4 

2171.6          £ 

2' 

4' 

5443-5 

7892.8 

2173-8         ^ 

4' 

6' 

5446.7 

7895  •  2 

2176.0              « 

6' 

8' 

5449-9 

7897-6 

2178.1   ^XTJ 

8' 

10' 

5453-0 

79OO  .O 

2180.3  ^:  ^ 

10' 

12' 

5456.1 

7902.4 

2182  .5  — 

12' 

14' 

5459-3 

7904.8 

2184  .6  .fc: 

14' 

16' 

5462.5 

7907.2 

2186.8  c& 

16' 

18' 

5465-7 

7909.7 

2189.0^: 

18' 

20' 

5468.9  10°? 

79I2.I 

2191  .2^^"] 

20' 

22' 

5472.  I  J:  : 

7914.5 

2193  .4  *d^  ^ 

22' 

24' 

5475-3- 

7916.9 

2195  .6  <j*  ^ 

24' 

26' 

5478.  5.  |:: 

79I9-3 

2197.8       •£ 

26' 

28' 

5481  .7  g 

7921.7 

22OO.O           <^ 

28' 

3o; 

5484.9  jir  : 

7924.1 

22O2  .  2           "£ 

30; 

5488.1  t£5 

7926.5 

2  2  04  .  4            Nt- 

32 

34' 

5491-3  £$£ 

7928.9 

22O6.6           ^ 

34' 

36' 

5494-  5  £,  . 

793L4 

2208.8           ^ 

36; 

38' 

5497-  7  «f  " 

7933-8 

2211  .O 

38' 

40' 

5500.9 

7936.2 

2213.3 

40' 

42' 

5504-1 

7938.6 

2215-5 

42' 

44' 

5507-3 

7941.0 

2217.7 

44' 

46' 

7943-4 

222O  .O 

46' 

48' 

55I3-7 

7945-8 

2222  .  2 

48' 

5°' 

5516.9 

7948.2 

2224.4 

5°' 

5^' 

5520.2 

7950.6 

2226  .6 

52^ 

54' 

5523  -4 

7953-o 

2228.8 

56' 

5526.6 

7955-4 

2231.1 

56' 

58' 

79^7.8 

2233  .3 

58' 

103 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

88° 

5533-0 

7960.2 

2235-5 

88C 

2' 

5536.3 

7962  .6 

2237-7 

2' 

4' 

5539-5 

7965.0 

2239.9 

4' 

6' 

5542.7 

7967.4 

2242  .  2 

6' 

8' 

5545-9 

7969.8 

2244.4                      H 

8/ 

10' 

5549-1 

7972.2 

2246.7      1 

10' 

12' 

5552.4 

7974-6 

2248.9     - 

12' 

14' 

5555-6 

7977-o 

22.51.2      .£ 

14' 

16' 

5558.9 

7979-4 

2253.4            03 

16' 

18' 

5562.1 

7981.8 

2255-7        £ 

18' 

20' 

5565-3 

7984.1 

2258.0        "5 

20-: 

22' 

5568.6        ? 

7986.5 

226o  .  2    ^  d/J 

22' 

24' 

5571-8          6 

7988.9 

2262  .  5  o:  j 

24' 

26' 

5575-1        ~ 

7991-3 

2264.7,, 

26' 

28' 

5578.3    4 

7993-7 

2267  .0  .h. 

28' 

30' 

5581.5    <£ 

7996.1 

2269.3^ 

30' 

32' 

5584.8    £ 

7998.5 

2271  .7  £: 

32' 

34' 

5588.0      5 

8000.9 

2273.8  £94 

34' 

36' 

5591-3  iAo£ 

8003.3 

2276  .1       -    • 

36' 

38' 

5594-5   6.  ^ 

8005.7 

2278.4?:  * 

38' 

40' 

5597-8-    < 

8008.0 

2280.7         | 

40' 

42' 

5601.  i.  gs 

8010  .4 

2283.0         ^ 

42' 

44' 

5604.3  w 

8012.8 

2285.2          % 

44' 

46' 

5607  .6  o: 

8015.2 

2287.5         X 

46' 

48' 

5610  .  8  u?oc  . 

8017.5 

2289.8          « 

48' 

50/ 

5614.1   £3^ 

8019.9 

2292  .  i        !§ 

50' 

52' 

8022  .3 

2294.4        < 

52' 

54' 

5620.  6  <"  ^ 

8024.  7 

2296.7 

54' 

56' 

5623-9     •§, 

8027  .  i 

2299  .  o 

56' 

58' 

5627.1         w 

8029.5 

2301  .2 

58' 

89° 

5630.4        «« 

8031.8 

2303-5 

89° 

2' 

5633-7         « 

8034.2 

2305-8 

2' 

4' 

5636.9        £ 

8036.6 

2308.  i  ^a4 

4' 

6' 

5640.2        ;§ 

8038.9 

2310.4    • 

6' 

8' 

5643-5        < 

8041  .3 

2312.  7^:  : 

8' 

10' 

5646.8 

8043.6 

2315-  o|:  : 

10' 

12' 

5650.1 

8046  .0 

2317-3  & 

12' 

14' 

5653-4 

8048.3 

23I9-6   £:  : 

14' 

16' 

5656-7 

8050.7 

2321.9"    MOO 

16' 

18' 

5660  .0 

8053.1 

2323.2     ->0cj 

i8r 

2O' 

5663.3 

8055-5 

2326.6  3:  : 

20' 

22' 

5666.6 

8057.8 

2328.9  < 

22' 

24' 

5669.9 

8060.  2 

2331.2 

24' 

26' 

5673-2 

8062  .5 

2333-5 

26' 

28' 

5676-5 

8064.  9 

2335-8 

28' 

104 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

89°  30' 

5679.8 

8067.3 

2338.2 

89°  30' 

32' 

5683.1 

8069.6 

2340.5 

32' 

34' 

5686.4  ^4 

8072  .0 

2342.8 

34' 

36' 

5689.8 

8074.4 

2345.1     •    •    . 

36' 

38' 

5693.  I  ^   = 

8076.8 

2347-4";      - 

38' 

40' 

5696.4'^  ^ 

8079  .  2 

2349-8^-   = 

40' 

42' 

5699.8  '5r  : 

8081.6 

2352.1^ 

42' 

44' 

5703  -1  fi 

8083.9 

2354.  4  -a-  - 

44' 

46' 

57°6.  4  ^"0"N 

8086.3 

2356.8^ 

46' 

48' 

5709-7  «£io 

8088.7 

2359.  i  ^:  : 

48' 

50' 

5713-0^^ 

8091  .0 

vO    w    c. 
2361.5    chocs' 

50' 

52' 

5716.45-  = 

8093.4 

2363-8,3 

52' 

54' 

57I9-7 

8095-7 

2366  .  2  ^:    : 

54' 

56' 

5723-0 

8098.1 

2368.5 

56' 

58' 

5726.3 

8100  .  5 

2370.9 

58' 

9O° 

5729.6 

8102  .9 

2373-3 

9O° 

2' 

5732.9 

8105.2 

2375-7 

2' 

4' 

5736.3          ^ 

8107  .6 

2378.1 

4' 

v 

5739-6 

8109  .  9 

2380.4          4 

6' 

8' 

5743-o        | 

8112.3 

2382.8          - 

87 

10' 

5746.3        *£ 

8114.6 

2385.1         ! 

10' 

12' 

5749-7        ;g 

8117  .0 

2387-5          £ 

12' 

14' 

5753-0        ^ 

8119.3 

2389-8         a 

14' 

16' 

5756.4 

8121  .7 

2392-2         ^ 

16' 

18' 

5759-7        ^ 

8124.0 

2394.6        - 

18' 

20' 

5763  .0  Ji<^^ 

8126.4 

2397-0   ^o£? 

20' 

22' 

5766.  4;g=  | 

8128.7 

2399-4,6,  5 

22' 

24' 

5769  .  7  *3      " 

8131.1 

2401.8^5    < 

24' 

26' 

5773-0-^ 

8i33-4 

2404.1  g_ 

26' 

28' 

5776.4^ 

8i35-8 

2406  .  5  £" 

28' 

30' 

5779-8fH 

8138.1 

2408  .9  ,0: 

30' 

32' 

5783-1  »Jt4 

8140.5 

2411.3^0  e,  ^ 

32' 

34' 

5786.5  «+6 

8142  .8 

2413.7    ^-   H 

34' 

36' 

5789.8?:  * 

8145.2 

2416.  i  5-  £ 

36' 

38' 

5793.2^    -g 

8i47-5 

2418.  5  <"  5 

38' 

40' 

5796.6        £ 

8149.8 

2420.9     -^ 

40' 

42' 

5800.0        £ 

8152.2 

2423-3      "£ 

42' 

44' 

5803.3        S 

8154-5 

2425-7      £ 

44' 

46' 

5806.7                £ 

8156.9 

2428.1 

46' 

48' 

58IO.I               £ 

8159.2 

2430.5      | 

48' 

5°' 

5813.5               ^ 

8161.6 

2432.9      5 

50' 

52' 

8163.9 

2435-3 

52' 

54'. 

5820.3 

8166.2 

2437-7 

54' 

56' 

5823.7 

8168.  6 

2440  .  i 

56' 

58' 

5827.0 

8170.9 

2442.5 

58' 

105 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

91° 

5830.4 

8173.3 

2445.0 

91° 

2' 

5833.8 

8175.6 

2447.4 

2' 

4' 

5837.2 

8178.0 

2449.8 

4; 

6' 

5840  .6 

8180.3 

2452.3 

8' 

5844.0 

8182.6 

2454.7           + 

8' 

10' 

5847-4 

8184.9 

2457-1 

ro' 

Mi' 

5850.8 

8187.2 

2459-5          ^ 

12' 

14' 

5854-2 

8189.6 

2462.0           E 

14' 

16' 

5857.6 

8191  .9 

2464.4         £ 

16' 

18' 

5861.0 

8194.3 

2466.8          £ 

18' 

ao' 

5864.5  «*c>4 

8196.6 

2469.3 

20' 

22' 

5867.9  0- 

8199  .0 

2471.7    ^o£ 

227 

24' 

5871  -3  £:  ~ 

8201  .3 

2474-2  ,°:  5 

24' 

26' 

5874.8-3 

8203.6 

2476  .  6  ^    < 

26' 

28' 

5878.  2  |T    = 

8205  .9 

2479.1  |: 

28' 

30/ 

58Si.  6  *:: 

8208.2 

2481.5^ 

30' 

32' 

5885  .  O  £  MOO 

8210.6 

2484.  o«£- 

32' 

34' 

5888.5    jjr£tj 

8212  .9 

2486  .  4  ^  "?  • 

34' 

36' 

5892  .O    N  "*«- 

8215.2 

2488.9  "^  - 

36' 

38' 

5895-4|:  : 

8217.5 

2491.3?:  ,° 

38' 

40' 

5898.8 

8219.8 

2493-8         g 

40' 

42' 

5902.3 

8222  .  i 

2496  .  2         'a 

42' 

44' 

5905-7 

8224.5 

2498.7         ^ 

44' 

46' 

5909.2 

8226.8 

2501.1         £ 

46' 

48' 

5912  .6 

8229  .  i 

2503-6         ^ 

48' 

So' 

5916.0 

8231.4 

2506.1         ^ 

5o; 

52' 

59*9-5 

8233.8 

2508.6        ^ 

54' 

5922.9 

8236.1 

2511  .  o 

54' 

56' 

5926.4 

8238.4 

2513  .  5 

56' 

58' 

5929.8 

8240.7 

2516  .0 

58' 

92° 

5933-2 

8243.0 

2518.5 

92° 

2' 

5936.7  ^±4 

8245.3 

2521  .  o 

2' 

4' 

5940.1    . 

8247.7 

2523-5  "^ 

4' 

6' 

5943.  6  ,g:  : 

8250.0 

2526  .06,- 

6' 

8' 

5947  -o-g 

8252.3 

2528.5^  " 

8' 

10' 

5950-5  |:  : 

8254.6 

2531.  o|:   : 

10' 

12' 

5954-0  £_  . 

8256.9 

2533-5^ 

12' 

14' 

5957-  4  xVo 

8259.2 

2536.0^:   = 

14' 

16' 

5960  .  9  oi  t^vd 

8261.5 

2538.5   gtt 

16' 

18' 

5964.3  £$£ 

8263.8 

2541.0     4« 

18' 

20' 

5967.  8  |=  = 

8266.1 

2543-  5  5:  : 

20X 

22' 

5971-3 

8268.4 

2546  .0 

22' 

24' 

5974-8 

8270.8 

2548.5 

24' 

26' 

5978.2 

8273.1 

2551-1 

26' 

28' 

598i.7 

8275.4 

2553  -6 

28' 

106 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


i     Whole 
f      Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

92°  30' 

5985-2 

8277.7 

2556.1 

92°  30' 

3*' 

5988.7 

8280.0 

2558.6 

32' 

34' 

5992.2 

8282.3 

2561  .  I 

34' 

36' 

5995-7  ^*  + 

8284.6 

2563.7 

36' 

38' 

5999-1  *e  ^ 

8286.9 

2566.2 

38' 

40' 

6002  .6  5" 

8289.2 

2568.7 

40' 

42' 

6006.  i  15. 

8291.5 

2571.2 

42' 

44' 

6009.  6  £"  ' 

8293.8 

2573-8 

44' 

46' 

6013.1  g.  . 

8296.  I 

2576.3 

46' 

48' 

6oi6.6^H 

8298.4 

2578.9 

48' 

5o' 

6020.1  SjS'S 

8300.  7 

2581.4 

5°; 

5*' 

6023.6^" 

8303.0 

2584.0          4 

52 

54' 

6027.  i  j:  : 

8305-3- 

2586.5          6 

54 

56' 

6030.6 

8307.6 

2589.1         X 

56' 

58' 

6034.1 

8309.9 

2591.6         "g 

58' 

93° 

6037.7 

8312.2 

2594.1         to 

93° 

2' 

6041  .2 

8314-5 

2596.7          % 

2' 

4' 

6044.7 

8316.8 

2599.2               vo 

4' 

6' 

6048.3 

8319.1 

2601.8  ^£% 

6' 

8' 

6051.8 

8321.4 

2604.4^0-  ;g 

8' 

10' 

6055.3 

8323-7 

2606.  9?    < 

10' 

12' 

6058.9 

8326.0 

2609.  5.  £: 

12' 

14' 

6052  .4 

8328.3 

26l2  .0  CO 

14' 

16' 

6056.0 

8330.6 

2614.6  ,§: 

16' 

18' 

6059.5 

8332.9 

2617  .2    «?•?    . 

1  8' 

.    .  •<$• 

CO  O   ^ 

20' 

6073.0  ^c-" 

8335.1 

2619.8^  w: 

20' 

22' 

6076.65°:  : 

8337-4 

2622  .4^-  S5 

22' 

24' 

6080.    I    *-< 

8339-7 

2625.0          -g 

24' 

26' 

6083.7-*=  = 

8342.0 

2627.5     -a 

26' 

28' 

6087.2  co 

8344.2 

2630.1      <« 

28' 

30' 

6090.  7£:  ' 

8346.5 

2632.7     ^ 

30' 

32' 

6094.3  «"«:« 

8348.8 

2635.3        y 

32' 

34' 

6097.8  £3£ 

8351-1 

2637.9     ^ 

34' 

36' 

6ioi.4^:  , 

8353-3 

2640.4      3j 

36' 

38' 

6104.9  ^ 

8355.6 

2643.0 

38' 

4o' 

6108.5 

8357-9 

2645  -6 

40' 

42' 

6112  .0 

8360.2 

2648.2 

42' 

44' 

6115  .6 

8362.4 

2650.8 

44' 

46' 

6119.2 

8364.7 

2653.4 

46' 

48' 

6122  .  7 

8367.0 

2656.0 

48' 

50' 

6126  .  3 

8369-3 

2658.6 

50' 

52' 

6129.9 

8371.6 

266l  .  2 

52' 

54'- 

6i33-4 

8373-8 

2663.8 

54' 

56' 

6137.0 

8376.1 

2666.4 

56' 

58' 

6140  .  6 

8378.4 

266Q  .O 

58' 

107 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord  . 

External 
Secant. 

Whole 
Angle. 

94° 

6144.  2 

8380.7 

2671  .6 

94° 

2' 

6147.8 

8383-0 

2674.2 

2' 

4' 

6151  .4 

8385.2 

2676.8 

4' 

6' 

6I55-0 

8387.5 

2679.5 

6' 

8' 

6158.6 

8389.8 

2682.1          4 

8' 

10' 

6162  .  I 

8392.1 

2684.7         6 

10' 

12' 

6165.7 

8394.3 

2687.3        ^ 

12' 

14' 

6169.3 

8396.6 

2690.0         £ 

14' 

16' 

6172  .9    ... 

8398.9 

2692.6        (g 

16' 

18' 

6176.5  ^°:r 

8401  .  I 

2695.3        js 

18' 

20' 

6180.  i  £:  = 

8403.4 

2697.9        ^ 

20' 

22' 

6183.8-3 

8405  .  7 

2700  .6  *>  <>£ 

22' 

24' 

6187.  4  'E:  = 

8408.0 

2703.  2|:  £ 

24' 

26' 

6191  .0  ^ 

8410.  2 

2705-9-3     < 

26' 

28' 

6194.6  £:  = 

8412.5 

2708.5  .£- 

28' 

30' 

6198.  2    w  rLvcJ 

8414-7 

2711  .  2  ^ 

30/ 

32' 

6201  .9    £^£ 

8417  .0 

2713.  9-^: 

32' 

34' 

6205.5^  : 

8419.3 

2716.5^ 

34' 

36' 

6209.  I  < 

8421.5 

2719.  2^^"". 

36' 

38' 

6212  .  7 

8423.8 

2721  .8  ?:  .§ 

38' 

40' 

6216.3 

8426  .  o 

2724.5          *g 

40' 

42' 

6220.0 

8428.3 

2727.2         j?- 

42' 

44' 

6223.6          4 

8430.5 

2729.8      "£ 

44' 

46' 

6227.3 

8432.8 

2732.5      ^ 

46' 

48' 

6230.9         £ 

8435-0 

2735.2          4 

48' 

5o' 

6234.5         I? 

8437-3 

2737-9        ^ 

5o; 

52' 

6238.2         'a 

8439-5 

2740.6         < 

54' 

6241.8  ^£ 

8441.8 

2743-3 

54' 

56' 

6245-5  6-°^ 

8444.  o 

2746.0 

56' 

58' 

6249.  i  £"    ^ 

8446.3 

2748.6          ^ 

58' 

95° 

6252-7!:  £ 

8448.6 

275J-3         ^ 

95° 

2' 

6256.  4£    ^ 

8450.9 

2754-0    .  .;*: 

2' 

4' 

6260  .0  £. 

8453-1 

2756-7  "'a^ 

4' 

6' 

6263.7^4 

8455-4 

2759.4^:  'a 

6' 

8' 

6267.3   ££~ 

8457-6 

2762.  i^  "£ 

8' 

10' 

6271.0^     £ 

8459-8 

2764.8.?:  «2 

10' 

•12' 

6274.7^  -3 

8462  .  1 

2767-5^     j 

I27 

14' 

6278.3         a 

8464.3 

277°.  2  £'  -c 

14' 

16' 

6282.0        ™ 

8466.6 

2772.9^-5 

16' 

18' 

6285.7         £ 

8468.8 

2775.6        M 

18' 

id 

20' 

6289.4            *;• 

8471.1 

20' 

22' 

6293.1 

8473-3 

2781.0 

22' 

24' 

6296.7 

8475-6 

2783-7 

24' 

36' 

6300.4 

8477-8 

2786.5 

26' 

28'  '  6304.1 

8480.  I 

2789.2 

28' 

108 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

95°  3*>' 

6307.8 

8482.3 

2791.9 

95°  30' 

32' 

63II-5 

8484.6 

2794.6 

32' 

34' 

6315.2 

8486.8 

2797.4 

34' 

36' 

6318.9  ^ds4 

8489.1 

2800.1     .    .   . 

36' 

38' 

6322  .6  6 

8491.3 

28O2    .9      *°   ^M 

38' 

40' 

6326.3  2" 

8493-5 

2805.  6£=  = 

40' 

42' 

6330.0  §-  - 

8495-8 

2808.3-3 

42' 

44' 

6333-  7  •£'  ' 

8498.0 

2811.1-5,=  = 

44' 

46' 

6337.4  fc:  : 

8500.  2 

2813.9^ 

46' 

48' 

634L  I  £00  « 

8502  .4 

2816.6^-  : 

48' 

5°' 

6344-8  B»£ 

8504.6 

CO   t-  0 
28l9  .4     rOO    4 

50' 

52' 

6348.5-0 

8506.8 

2822  .  2  ^  M 

52' 

54' 

6352.  2  5=    =, 

8509  .0 

2825.  ojp  = 

56' 

6356.0 

85II.3 

2827.7 

56' 

58' 

6359.7 

8513.5 

2830.5 

58' 

96° 

6363.4 

8515-8 

2833.2 

96° 

2' 

6367.1 

8518.0 

2836.0 

2' 

4' 

6370.8        64 

8520.3 

2838.7 

4' 

6' 

6374-6     .£  - 

8522.5 

2841.5 

6' 

8' 

6378.3       I*  ;° 

8524.8 

2844.3 

8' 

10' 

6382.0     £-5 

8527.0 

2847.1 

10' 

12' 

6385-7    £•& 

2849.9 

12' 

14' 

6389-5    ^ 

853I-5 

2852.7 

14' 

16' 

6393-2    ££ 

8533-7 

2855.4 

16'  . 

18' 

6397-0   ^2 

8535-9 

2858.2 

18' 

20' 

6400.7  «?;§£• 

8538.1 

2861  .0  ""** 

20' 

22' 

6404-  5  |    | 

8540.3 

2863.8  6,  . 

22' 

24' 

6408.3^    < 

8542.5 

2866.6^    " 

24' 

26' 

6412  .0  •% 

8544.8 

2869.4|:  . 

26' 

28' 

6415.8^ 

8547-0 

2872.2'^" 

28' 

30' 

6419-5! 

8549.2 

2875.0^:  s 

3°' 

32' 

6423-3  «      - 

8551  .4 

2877  .8  coco  ro 

32' 

34' 

6427.  o£    2 

8553-6 

2880.6    "23 

34' 

36' 

6430.8^     6 

8555-9 

2883.4?-    . 

36' 

38' 

6434.  6  <  <>2 

8558.1 

2886.  2  <J"    " 

38' 

40' 

6438.3  &! 

8560.3 

2889.0 

40' 

42' 

6442  .  i     13^ 

8562.5 

2891.8 

42' 

44' 

6445-8     ftg 

8564.7 

2894.6 

44' 

46' 

6449  .6    /;  "? 

8567.0 

2897.4 

46' 

48' 

6453-4   ^H 

8569.2 

2900.  2 

48' 

5°' 

6457-2    ^3 

8571  .4 

2903.1 

5°' 

52' 

6461  .0      "*<! 

8573.6 

2905.9 

52' 

54' 

6464.8     ? 

8575-8 

2908  .  7 

54' 

56' 

6468.6     < 

8578.0 

2911  .6 

56' 

58' 

6472.4 

8580.2 

2914.4 

58' 

109 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

97° 

6476.  I 

8582  .4 

2917.3 

97° 

2' 

6479.9 

8584.6 

292O  .  2 

2' 

4' 

6483.7 

8586.8 

2923.0 

4' 

6' 

6487.5          4 

8589.0 

2925.9 

6' 

8' 

6491.3          * 

8591.2 

2928.7          4 

8' 

10' 

6495.1         fc 

8593  •  5 

2931  .6 

10' 

12' 

6498.9          fi 

8595.7 

2934.5        * 

12' 

14' 

6502.7         £ 

8597-9 

2937-3        8 

14' 

16' 

8600  .  i 

2940.2       £ 

16' 

18' 

6510.4         ^ 

8602.3 

2943-0        £ 

18' 

20' 

6514.2  .  .  r: 

8604.5 

2945-9       ^ 

20' 

22' 

6518.0  £*£ 

8606.7 

2948.8  ^<>? 

22' 

24' 

6521  .9  £r  5; 

8608.9 

2951  .  7  o-  ;g 

24' 

26' 

6525-7^ 

8611.1 

'2954.5-3    < 

26' 

28' 

6529.  5  £= 

8613.3 

2957.  4.  g: 

28' 

30' 

6533.3  fc. 

8615.5 

2960.3  ^ 

3O/ 

32 

6537-2*^    . 

8617.7 

2963.2  £  = 

32' 

34 

6541  .0    woo  ? 

8619  .  9 

2966.  1  ^^4 

34' 

36' 

6544.8  "So 

8622.1 

2968.9  ^'°  M 

36; 

38' 

6548.7?:  5 

8624.3 

2971.  8|;     | 

40' 

^2 

6552.5     -g 

8626.5 

<    ^ 

2974.7      g 

40' 

42' 

6556-4        <£ 

8628.7 

2977.6       -ft 

42' 

44' 

6560.2          £ 

8630.8 

2980.5       r£ 

44' 

46' 

6564.0           °° 

8633.0 

2983.4       -2 

46' 

48' 

6567.9                          M 

8635-2 

2986.3        ^ 

48' 

50' 

6571.8      5 

8637.4 

2989.2       -d 

5°' 

52' 

6575.7     < 

8639.6 

2992.1       ^ 

$2f 

54' 

6579.5 

8641  .8 

2995.0 

54' 

56 

6583.4 

8644.0 

2997.9 

56' 

58' 

6587.2 

8646.2 

3000  .  8 

58' 

98° 

6591.1 

8648.4 

3003.8 

98° 

2' 

6595.0  ^4 

8650.6 

3006.7    . 

2' 

4' 

6598.8    .      - 

8652.8 

3009  .  6  ^<*-* 

4' 

6' 

6602  .  7  £:  = 

8655.0 

3012.5  ^6,  . 

6' 

8' 

6606.6-3 

8657.2 

8' 

10' 

6610.5  c|:  ^ 

8659.3 

30i8.  4  |:  : 

10' 

12' 

6614.4  ^ 

8661.5 

3021.3^ 

12' 

14' 

6618.3  •*-" 

8663.7 

3024.2  ,0:  : 

14' 

16' 

6622.2      N^cS 

8665.8 

3027.2   £9  ^ 

16' 

18' 

6626.1  «££ 

8668.0 

3030.1       ^M    Jf 

18' 

20' 

6630.05:  : 

8670.2 

TJ 
3033.    I    ^      , 

20' 

22' 

6633.9^ 

8672.4 

3036.0 

22' 

24' 

6637.8 

8674.5 

3039.0 

24' 

26' 

6641  .  7 

8676.7 

3042.0 

26' 

28' 

6645.6 

8678.9 

3044.9 

28' 

110 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


i       Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

98°  30' 

6649.5 

868I.I 

3047.9 

98°  30' 

32' 

6653-5 

8683.3 

3050.9 

32' 

34' 

6657.4 

8685.5 

3053.9 

34' 

36' 

6661.3  u^4 

8687.6 

3056.8    .    .   . 

36' 

38' 

666s  .  2    .     H 

^P:   : 

8689.8 

3059.8^? 

38' 

40' 

6669.  I  g" 

86,92  .0 

3062  .8  ^:  : 

40' 

42' 

6673  .  i  2. 

8694  .  2 

3065.8-3 

42' 

44' 

6677.  og:  : 

8696.3 

3068.8-^  = 

44' 

46' 

6681.0  g.  . 

8698.5 

3071.  8<2 

46' 

48' 

6684  .  9  *£  V"w 

8700.  7 

3074.7.0:    5 

48' 

50' 

6688.8  S'S'S 

8702  .9 

Ov  O  00 

3077.7   «n4 

50' 

52' 

6692.8,5* 

8705  .0 

3080.7  ^  H 

52' 

54' 

6696.7^  = 

8707  .  2 

3083.75=  = 

54' 

56' 

6700  .6 

8709.3 

3086.7 

56' 

58' 

6704  .  6 

87H.5 

3089.7 

58' 

99° 

6708.5 

8713.7 

3092.7 

99° 

2' 

6712.5 

8715.9 

3095.7 

2' 

4' 

6716.5 

8718.0 

3098.7 

4' 

6' 

6720  .4 

8720  .2 

3IOI.7          ^ 

6' 

8' 

6724.4        £ 

8722.3 

3104.8 

8' 

10' 

6728.3        -g 

8724.5 

3107.8         £ 

10' 

12' 

6732.3      -a 

8726.6 

3110.8         *g 

12' 

I4/ 

6736.3      <£ 

8728.8 

3113-8         '2, 

14' 

16' 

6740.2        «S 

8730.9 

3H6.9         ^ 

16' 

18' 

6744-2      ^ 

8733.1 

3119.9         £ 

1  8' 

20' 

6748.  I    "'•><>£ 

8735.3 

3122.9  +£« 

20' 

22' 

6752.  l|:  | 

8737.4 

3126  .0  o,  »c 

22' 

24' 

6756.1-3     * 

8739.6 

3129.0^"  ^ 

24' 

26' 

8741.7 

3132.0  £^ 

26' 

28' 

6764.0  w 

8743.9 

3I35«I  '$" 

28' 

30' 

6768.  o|r 

8746.0 

3138.1^: 

3°' 

32' 

6772  .0  3><»  4 

8748.2 

3141    .1       OvK       . 

32' 

34' 

6776.0  £3M 

8750.3 

3144.2       ^^    M 

34' 

36' 

6780.0  5.  £ 

8752.5 

3147.25.    6 

36' 

38' 

6784.  o<T  -g 

8754.7 

3150.  3<~  2 

38' 

40' 

6788.0        £ 

8756.8 

3153.4        •§. 

40' 

42' 

6792.0         £ 

8759.0 

3156.4     «2 

42' 

44' 

6796.1         ^ 

8761.1 

3I59-5       <2 

44' 

46' 

68OO  .  I            oc 

8763.3 

3162.6         -: 

46' 

48' 

6804.1            £• 

8765.4 

3165.6        ^ 

48' 

50' 

6808.  I           | 

8767.5 

3168.7        | 

50' 

52' 

68l2.  I 

8769.7 

3171.8 

52' 

54' 

68l6.  I 

8771.8 

3174.8 

54' 

56' 

6820.2 

8774.0 

3J77-9 

56' 

58' 

6824.2 

8776.1 

3181  .0 

58' 

111 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

1OO° 

6828.2 

8778.2 

3184.1 

1OO° 

2' 

6832.2 

8780.3 

3187.2 

2' 

4' 

6836.3 

8782.5 

3190.3 

4' 

6' 

6840.3 

8784.6 

3193.4 

6' 

8' 

6844  .4 

8786.8 

3196.5           ? 

8' 

10' 

6848.4 

8789.0 

3199.6         | 

ior 

12' 

6852.5 

8791.1 

3202.7         -3 

I27 

14' 

6856.5 

8793.3 

3205.8         -g 

14' 

16' 

6860.6  "&  + 

8795.4 

3208.9         w 

16' 

18' 

6864.  7   6  6- 

8797.6 

3212.0         £ 

18' 

to] 

6868.  7  H 

8799.7 

3215.1          % 

20' 

22' 

8801.8 

3218.2       ^ 

*    22' 

24' 

6876.  9  ££ 

8803.9 

3221.3  jig 

24' 

26' 

6881.  o||= 

8806.  I 

3224.5   -3 

26' 

28' 

6885.0   o  ^ 

J              roc/300 

88o8.2 

3227.6  .£ 

28' 

3o' 

6889.1   £$£ 

8810.3 

3230.8     <£ 

30' 

32' 

6893.2?,  - 

8812.4 

3233.9     «2 

32' 

34' 

6897.  3<f 

8814.6 

3237.1      ^  . 

34' 

36' 

6901.3 

8816.7 

3240.2  ,^  ? 

36' 

38' 

6905.4 

8818.  8 

3243.4^|| 

38' 

40' 

6909.5 

8821.0 

3246.5^   i 

40' 

42' 

6913.6         4 

8823.1 

3249.6-^  -^ 

42' 

44' 

6917.7 

8825  .2 

3252  .80:    & 

44' 

46' 

6921.8         j| 

8827.4 

3255.9.0  ^ 

46' 

48 

6925.9        ^ 

8829.5 

3259.1    o       t 

48' 

50' 

6930.0      |, 

8831.6 

3262.3^^ 

50' 

52' 

6934.1      <£ 

8833.7 

52' 

54' 

6938.2    .  ;2 

8835-8 

3268,'6^-g^ 

54' 

56' 

6942.3  !?4<2 

8838.0 

3271-7     '§, 

56' 

58' 

6946.4^:    £ 

8840.1 

3274-9      g 

58' 

101° 

/;              /:  ^     ^ 
6950.6  g^  *d 

8842.2 

3278.1     X 

2' 

6954.8$-  r 

8844-3 

3281.2                M 

2' 

4' 

6958.9  £- 

8846.4 

3284.4     -c? 

4' 

6' 

6963-0^   . 

88^8.6 

3287.6  ?6 

6' 

8' 

6967.1       ^06H 

88^0.7 

3290.8       £ 

8' 

10' 

697I-2^.*5§ 

8852.8 

3294.0       -^ 

ior 

12' 

6975  .4  2~    *e3 

8854.9 

3297.2        n 

12' 

14' 

6979.5            '§, 

8857.0 

3300.4       ^ 

14' 

16' 

6983.7            ^ 

8859.2 

3303.6        ^ 

16' 

18' 

6987.8            ^ 

8861.3 

3306.8 

18' 

20' 

6991.9             | 

8863.4 

3310.1        % 

20' 

22' 

6996  .  I             *- 

8865.5 

3313  .3 

22' 

24' 

7OOO.2            3 

8867.6 

3316.6 

24' 

26' 

7004.4 

8869.7 

33I9.8 

26' 

28' 

7008.5 

8871.8 

3323-0 

28' 

110 


IX.— FUNCTIONS  OP  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

101°  30' 

7012.7 

8873-9 

3326.2 

10103o/ 

3^' 

7016  .  9 

8876.0 

3329.4 

32A 

34' 

7021  .0 

8878.1 

3332.7 

34' 

36' 

7025.2    .  .   . 

8880.3 

3335-9    .  .  . 

36' 

38' 

7029.4  "V*? 

8882.4 

3339-2  ^-M 

38' 

40' 

7°33-6|:  : 

8884.5 

3342.4^-  - 

40' 

42' 

7037-8^ 

8886.6 

3345.  6-g 

42' 

44' 

7042.0-^  : 

8888.7 

44' 

46' 

7046  .  I  ^ 

8890.8 

3352  .  i  w 

46' 

48' 

7050.3  ..3:  = 

8892  .9 

3355  -4  5:  : 

48' 

5°' 

7054.5  £«|2> 

8895.0 

3358.6  4--S 

50' 

??.'„; 

7058.7  SWg. 

8897.1 

3361  .8  ^ 

52' 

54' 

7062  -9  ?;    : 

8899.2 

3365.  !?:    : 

54' 

56' 

7067.!^ 

8901.3 

3368.3 

56' 

58' 

7071.3 

8903.4 

3371-6 

58' 

102° 

7075-5 

8905-5 

3374-9 

1O2° 

2' 

7079.8 

8907  .6 

3378.2 

2' 

4' 

7084.0 

8909.7 

4' 

6' 

7088.2' 

8911  .8 

3384.7 

6' 

8' 

7092.4 

8913.9 

3388.0 

8' 

10' 

7096  .6 

8916  ,o 

339J-3 

TO' 

12' 

7100  .9 

8918.1 

3394-6 

12' 

14' 

7105.1 

8920.2 

3397-9 

14' 

16' 

8922.3 

3401.2 

16' 

18' 

7"3-5  ^^ 

8924.4 

3404-5 

18' 

20' 

7117.7  ^°" 

8926  .4 

3407.8  «^»  j 

20' 

22' 

7122  .0  ,°-  " 

8928.5 

34II-  I   6.  j 

22' 

24' 

7126.2  -3 

8930.6 

3414.4^' 

24' 

26' 

7130.  4-|:  : 

8932.7 

34I7-7  fi.  - 

26' 

28' 

7134.7  in 

8934.8 

342i.  o|/  * 

28' 

30' 

7138.9^ 

8936,9 

3424.3,0"  = 

30' 

32' 

7143.2   ^  & 

8939.0 

3427.6  o-.o 

32 

34' 

7147.4  ££;: 

8941  .  i 

3431.0  ^H1^ 

34; 

36' 

7151  .72.. 

8943.2 

3434.3?.  : 

38' 

7156.  o<5"  : 

8945-3 

3437  -6  < 

38' 

40' 

7160.2 

8947-3 

3440.9 

40' 

42' 

7l64-5 

8949.4 

3444-2 

42' 

44' 

7168.7 

895I-5 

3447-6 

44' 

46' 

8953.6 

3450.9 

46' 

48' 

7177.3 

8955-7 

3454.2 

48' 

56* 

7181.6 

8957.7 

3457-6 

50' 

52' 

7185.9 

8959.8 

3460.9 

52 

54' 

7190.2 

8961.8 

3464.3 

54! 

56' 

7*94-5 

8963.9 

3467.6 

56 

58' 

7198.8 

8966.0 

3471-0 

58' 

113 


IX.— FUNCTIONS  OP  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 

Chord. 

External 
Secant. 

Whole 
Angle. 

103° 

7203.1 

8968.1 

3474-4 

IO3° 

2' 

7207.4 

8970.2 

3477-8 

2' 

4' 

7211.7 

8972  .  2 

348i  .2 

4' 

6' 

7216  .0 

8974.3 

3484.5 

6' 

8' 

7220.3      ^~ 

8976.4 

3487.9         -* 

a' 

10' 

7224.6      £: 

8978.4 

3491-3         6 

10' 

12' 

7229  .O  .  ^ 

8980.5 

3494-7        3 

12' 

14' 

?_ 

8982.5 

3498.1         § 

14' 

16' 

7237.6     w 

8984.6 

3501.5      £ 

16' 

1  8' 

7241.9      ,0  = 

8986.7 

3504.9        | 

i8r 

20' 

7246.2    a  2 

8988.7 

3508.3 

20'' 

22' 

7250.6    ££ 

8990.8 

3511.7  ^dN 

22' 

24' 

7254.9   ^§: 

8992  .8 

3515  .  I    <2;    ^§ 

24' 

26' 

7259.2   <f 

8994.9 

3518  .  5  J    *3 

26' 

28' 

7263.6 

8997.0 

35?*  -9.fi 

28' 

30' 

7267.9 

8999  .0 

3525.3^ 

30' 

32' 

7272.3 

9001  .  i 

3528  7  ^: 

32 

34 

7276.6  ^4 

9003.2 

3532-2  ^4 

34 

36' 

7281  .0 

9OO5  .  2 

3535  -6  TJ~  : 

36' 

38' 

7285-3*-  - 

9007.3 

38' 

40' 

7289.7^ 

9009.3 

3542-5 

«t 

40' 

42' 

7294.0  'a-  - 

9011  .4 

3545-9 

'E 

42' 

44' 

7298.4  ^ 

9013.4 

3549-4 

^ 

44' 

46' 

7302.  8  -:H^ 

9OI5-5 

3552-8 

o 

46' 

48' 

7307.2  codd 

3556.  3( 

0 

48' 

5°' 

7311.6^- 

9019  .6 

3559-7' 

J 

5°' 

52' 

7316.  Ojj:  : 

9021  .6 

3563-1 

<^ 

52/, 

54' 

7320.4 

9023-7 

3566.6 

54 

56' 

7324.8 

9025.7 

3570-0 

56' 

58' 

7329.2 

9027  .8 

3573-5 

58' 

1O4° 

7333-5 

9029.9 

3577-0 

1O4° 

2' 

7337-9      a4 

9031.9 

3580  .  5 

2' 

4' 

7342.3       •  M 

9034.0   j   3583.9 

i^  0  ^> 

4' 

6' 

7346.7     £: 

9036  .0 

3587-4 

6-  : 

6' 

8' 

7351-1*  g 

9038,1 

3590.9^ 

8' 

10' 

7355-5     '£ 

9040  .  2 

3594.3-^  : 

10' 

12' 

7360.0     i-^ 

9042  .2 

3597-8/2 

I2X 

14' 

7364.4     £'H 

9044.3 

3601  .21,0-   = 

14' 

1  6' 

7368.8       do' 

9046.3 

3604.  7h^ 

16' 

18' 

7373-2      3" 

9048  .4 

3608.  2p:  >j 

i8r 

20' 

7377-6     |- 

9050.4 

3611.71!'  : 

20' 

22' 

7382.1 

9052.5 

3615-2] 

22' 

24' 

7386.5 

9054.5 

3618.7 

24' 

26' 

7390.9 

9056  .6 

3622.2 

26' 

28'    7395-4 

9058.6       3625.7 

28' 

114 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

lO4°3o' 

7399-8 

9060.6 

3629.2 

lO4°3o' 

32' 

7404.3 

9062  .  7 

3632.7 

32' 

34' 
36' 

7408.8 
7413-2    .  .  . 

9064.7 
9066.8 

3636-2     .    .    . 
3639.7   ">*  + 

34' 
36' 

38' 

7417-7  "°^ 

9068.8 

3643-3^6,  = 

38' 

4o' 

7422.1  £-  - 

9070.8    3646.8-3 

40' 

42' 

7426.613 

9072.9  13650.3-5,;  : 

42' 

44* 

743*.  O-g:  = 

9074.9  13653-9^ 

44' 

46' 

7435-  5*2 

9076.9     3657.4^  = 

46' 

48' 

7440.  0<o:  : 

9079  .0    3661  .  o  i  'r.1? 

48' 

5o; 

7444.5    £Sd 

9081  .0 

3664.  5  ^MW 

50' 

7449-0  £*£ 

9083.0 

3668.1  ^  = 

52' 

11' 

7453-5?:   : 

9085  .  i 

3671.6 

54' 

56' 

7458.  o< 

9087  .  i 

3675-2 

56' 

58' 

7462.5 

9089  .  i 

3678.8 

58' 

1O5° 

7467.0 

9091  .  i 

3682.3 

105° 

2' 

7471-5 

9093.2     3685.8 

2' 

4' 

7476.0 

9095.2     3689.4 

4' 

6' 

7480.5 

9097.2    3693-0 

6' 

8' 

7485.0 

9099.3    3696.6 

.         8' 

10' 

7489.5 

9101  .  3     3700  .  2 

10' 

I2f 

7494-0 

9103.3  3703.8 

12' 

14' 

7498.6 

9105.3  3707-4 

14' 

16' 

7503.1 

9107.4    13711  .0 

1  6' 

18' 

7507.6    . 

9109  .4    3714  .  6 

1  8' 

20'       75I2.I    10C  ? 

9111.4     3718.2  «A  «*  j 

20' 

22f 

7516.7,0:  i 

9113.4    13721.8  d-  j 

22' 

24' 
26' 

7521.2.3 

7525-  8-£:: 

9H5.5    37*5-4  2 

9II7-5     3729-1  .&  : 

24' 

26' 

28' 

753o.3<2' 

28' 

30' 

7534-  8^:: 

9121.5 

3736.3   c=  : 

3°' 

32' 

7539-4  JFdi? 

9123.5 

3739-9^^ 

a«; 

34' 

7543-9  £52. 

9125  .6 

3743-6  *H^ 

34' 

36' 

7548.55.  . 

9127  .6 

3747-25.  ; 

36' 

38' 

7553-0  <T  ' 

9129.6 

3750.  8< 

38' 

40' 

7557-6 

9131.6 

3754-5 

.         4°' 

42' 

7562.1 

9I33-6 

3758.2 

42' 

44' 

7566.7 

9I35-6 

3761.8 

44' 

46' 

757L3 

9137.6 

3765-5 

46' 

48' 

7575-9 

9139.6 

3769-I 

48' 

5°' 

7580.5 

9141  .  6 

3772.7 

s°; 

52' 

7585-1 

9143.6 

3776.4 

54' 

7589-7 

9I45-7 

3780.0 

54' 

56' 

7594-3 

9147.7 

3783-7 

56' 

58' 

7598.  Q 

9149.7 

3787.4 

58' 

115 


IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord. 

External 
Secant. 

Whole 
Angle. 

1OO° 

7603.5 

9i5!-7 

3791.0 

1OO° 

2' 

7608.1 

9153.7 

3794-7 

2f 

4' 

7612  .7 

9I55-7 

3798.4 

4' 

6' 

7617.3 

9I57-7 

3802  .0 

6' 

8' 

7621     .9                   °M 

9I59-7 

3805.7 

8' 

10' 

7626.5      £  = 

9161.7 

3809.4 

10' 

I2f 

763LI      -3 

9163.7 

3813-1 

12' 

14' 

7635.7      £= 

9165.7 

3816.8 

14' 

16' 

7640.4      <£ 

9167.7 

3820.5   ^&* 

16' 

18' 

7645.1       .0  = 

9169.7 

3824.2   £  ^ 

18' 

20' 

7649.7    ,/n 

9171.7 

3827.9  *'  ' 

20.' 

22' 

7654.4    ^'3£. 

9I73-7 

3831.6  "2.  . 

22' 

24' 

7659.0  ^: 

9I75-7 

3835  -4  $"  ' 

24' 

26' 

7663.7  *£< 

9177.7 

3839-1    £-  - 

26' 

28' 

7668.  3  -a 

9179.7 

3842.8  ^"0"M 

28' 

30' 

7672.9  £ 

9181.7 

3846.5   4i^ 

30' 

32' 

7677.6  ^ 

9183.7 

3850.2  ^  . 

32' 

34' 

7682.2  £^2 

9185.7 

3854.0  < 

34' 

36' 

7686.9  ^6.    ' 

9187.7 

3857.7 

36' 

38' 

7691  .5  %K~ 

9189.7 

3861.5 

38' 

40' 

7696.2       .t: 

9191.7 

3865.2 

40' 

42' 

7700.9      c& 

9I93-7 

3869  .0 

42' 

44' 

7705.5      Jh 

9I95-7 

3872.7 

44' 

46' 

7710.2           VC    0 

9197.7 

3876.5 

46' 

48' 

7714.9           fg 

9199.7 

3880.2 

48' 

50/ 

7719.6          2^' 

9201  .6 

3884.0 

50' 

52' 

7724.3          SI' 

9203.6 

3887.8 

52' 

54' 

7729.0 

9205.6 

3891  .6 

56' 

7733-7 

9207  .6 

3895  .4    ... 

56' 

58' 

7738.4 

9209  .6 

3899.1  ly;c"- 

58' 

1O7° 

7743-1 

9211.5 

3902.9  £-  : 

107° 

2' 

7747-9    -   -4 

92I3-5 

3906.7  -g 

2' 

4' 

7752.6  ^?C(- 

9215-5 

3910.5  -a-  : 

4' 

6' 

7757-3  %'  - 

9217.5 

3914.3^ 

6' 

8' 

7762.0  -g 

9219.4 

3918.1  &-^ 

8' 

10' 

7766.7  '$."  * 

9221  .4 

3921  .9     4ej  i- 

10' 

12' 

777J-4  g.  - 

9223.4 

3925-7    ^ 

12' 

14' 

7776.2  -^ 

9225.4 

3929.5    4-    * 

14' 

16' 

7780.9      r^^,- 

9227.3 

3933-3 

16' 

18' 

7785.7  ^32. 

9229-3 

3937-1 

18' 

20' 

7790.4  ^= 

9231.3 

3941  .0 

20' 

22' 

7795-2 

9233.3 

3944-8 

22' 

24' 

7800.0 

9235-3 

3948.7 

24' 

26' 

7804.7 

9237.2 

3952.5 

26' 

28'   '7809.5 

9239.2 

3956.4 

28' 

116 


-FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


Whole 
Angle. 

Tangent 
Distance. 

Long 
Chord  . 

External 
Secant. 

Whole 
Angle. 

107°  30' 

7814.2 

9241.2 

3960.2 

!O7°so' 

3^' 

7819  .  o 

9243.2 

3964.0 

32' 

34' 

7823.7 

9245.1 

3967.9 

34' 

36' 

7828.5 

9247.1 

397L7     •    • 

36' 

38' 

7833.4 

9249.1 

3975-6  ^- 

38' 

40' 

7838.1 

9251.0 

3979-5  &  ' 

40' 

42' 

7842.9 

9253.0 

3983-3  "5 

42' 

44' 

7847.7 

9255.0 

3987.2   •£.--   = 

44' 

46' 

7852.5  .  . 

9256.9 

3991.1  w 

46' 

48' 

7857.3  "^ 

92^8.9 

3995-0  &^ 

48' 

5°' 

7862.1  £s  = 

9260  .  8 

3998.9  522 

50' 

5'' 

7866.9^      5 

9262  .8 

4002  .8  ^ 

52' 

54' 

787L7   J    ' 

9264  .8 

4006.7  **  = 

54' 

S6' 

7876.5*-- 

9266.  7 

4010.6 

56' 

58' 

7881.3^;; 

9268.7 

4014.5 

58' 

1O8° 

7886.1  zz« 

9270  .6 

4018.3 

108° 

2' 

7890.94;*- 

9272  .6 

4O22  .  2 

2' 

4' 

7895.85= 

9274.5 

4O26  .  I 

4' 

6' 

7900.6 

9276.5 

4030.0          4 

6' 

8' 

7905.4 

9278.4 

4033.9          6 

8' 

10' 

7910.2 

9280  .4 

4037-8         ^ 

10' 

12' 

791  ^  .0 

9282.3 

4041.7         •* 

12' 

14' 

7919.9 

9284.3 

4045.6             03 

14' 

16' 

7924.7 

9286.3 

4049.6         J3 

J6' 

18' 

7929.6 

92.88.2 

4053.5      « 

i8x 

20' 

7934.5 

9290  .  2 

4057.4  ^cf 

20' 

22' 

7939-3  +6  + 

9292  .  2 

4061  .4   6.  -c 

22' 

24' 

7944-2     . 

9294.  I 

4065.3^  < 

24' 

26' 

7949.1  gs  = 

9296  .  I 

4069.3  |, 

26' 

28' 

7954.0  -g 

9298.0 

4073-2  £" 

28' 

30' 

7958.9  '$•"  = 

9300  .  o 

4077.2  |^ 

30/ 

32' 

7963-8  £.  . 

9301  .9 

4081  .2     «?*?   . 

32' 

34' 

7968.7  -Vr- 

9303-Q 

4085  .2  "*!;  H 

34 

36' 

7973.  -6  £c,~ 

9305-8 

4089.1  ^:    d 

36' 

38' 

7978.5  4;**- 

9307.8 

4093.1  <    ^ 

38' 

40' 

7983.4  32  : 

9309.7 

4097  -1         '§, 

40' 

42' 

7988.3   ' 

93H.7 

4101  .1 

42' 

44' 

7993-2 

9313.6 

4105.1         £ 

44' 

46' 

7998.1 

9315-6 

4109  .  o 

46' 

48' 

8003  .0 

93I7-5 

4113.0 

48' 

50' 

8007  .9 

93I9-4 

4117.0        ^ 

50/ 

52' 

8012.8 

9321  -4 

4121  .0 

52/ 

54- 

8017.7 

9323.3 

4125.0 

54' 

56' 

8022  .6 

9325.2 

4129.0 

56' 

58' 

8027.5 

9327  •  J 

^I33-1 

58' 

UNIVERSITY   OF    CALIFORNIA 
LIBRARY 


This  is  the  date  on  which  this 
book  was  charged  out. 


APH  20  1912 
vj\ 

«-?     v 

o^ 

NOV26   t942 


[30m-6,'ll] 


06857 


